Publications of Douglas N. Arnold

The present paper extends the landscape theory pioneered in Filoche and Mayboroda (Proc. Natl. Acad. Sci. USA 109(37):14761-14766, 2012), Arnold et al. (Commun. Partial Differ. Equ. 44(11):1186-1216, 2019) and David et al. (Adv Math 390:107946, 2021) to the tight-binding Schrödinger operator on Z^d. In particular, we establish upper and lower bounds for the integrated density of states in terms of the counting function based upon the localization landscape.

Keywords: Integrated Density of States, Anderson models, tight-binding model, Landscape Law

Spectral function is a key tool for understanding the behavior of Bose-Einstein condensates of cold atoms in random potentials generated by a laser speckle. In this paper we introduce a new method for computing the spectral functions in disordered potentials. Using a combination of the Wigner-Weyl approach with the landscape theory, we build an approximation for the Wigner distributions of the eigenstates in the phase space and show its accuracy in all regimes, from the deep quantum regime to the intermediate and semiclassical. Based on this approximation, we devise a method to compute the spectral functions using only the landscape-based effective potential. The paper demonstrates the efficiency of the proposed approach for disordered potentials with various statistical properties without requiring any adjustable parameters.

Keywords: spectral function, localization landscape, speckle potential

We construct local projections into canonical finite element spaces that appear in the finite element exterior calculus. These projections are bounded in L² and commute with the exterior derivative.

Keywords: cochain projection, commuting projection, finite element exterior calculus

This paper is concerned with the derivation and properties of differential complexes arising from a variety of problems in differential equations, with applications in continuum mechanics, relativity, and other fields. We present a systematic procedure which, starting from well-understood differential complexes such as the de Rham complex, derives new complexes and deduces the properties of the new complexes from the old. We relate the cohomology of the derived complex to that of the input complexes and show that the new complex has closed ranges, and, consequently, satisfies a Hodge decomposition, Poincaré type inequalities, well-posed Hodge-Laplacian boundary value problems, regular decomposition, and compactness properties on general Lipschitz domains.

Keywords: Differential complex, Hilbert complex, de Rham complex, elasticity complex, BGG resolution, finite element exterior calculus

Recent work has proved the existence of bounds from above and below for the integrated density of states (IDOS) of the Schrödinger operator throughout the spectrum, called the landscape law. These bounds involve dimensional constants whose optimal values are yet to be determined. Here, we investigate the accuracy of the landscape law in 1D and 2D tight-binding Anderson models, with binary or uniform random distributions. We show, in particular, that in 1D, the IDOS can be approximated with high accuracy through a single formula involving a remarkably simple multiplicative energy shift. In 2D, the same idea applies but the prefactor has to be changed between the bottom and top parts of the spectrum.

Keywords: Integrated Density of States, Anderson models, tight-binding model, Landscape Law

We study the finite element approximation of the Kirchhoff plate equation on domains with curved boundaries using the Hellan-Herrmann-Johnson (HHJ) method. We prove optimal convergence on domains with piecewise C^k+1 boundary for k >=1 when using a parametric (curved) HHJ space. Computational results are given that demonstrate optimal convergence and how convergence degrades when curved triangles of insufficient polynomial degree are used. Moreover, we show that the lowest order HHJ method on a polygonal approximation of the disk does not succumb to the classic Babuska paradox, highlighting the geometrically non-conforming aspect of the HHJ method.

Keywords: Kirchhoff plate, simply-supported, parametric finite elements, mesh-dependent norms, geometric consistency error, Babuska paradox

We consider the Neumann boundary value problem for an elliptic operator L = -div A grad + V on a bounded bi-Lipschitz domain in R^n. More generally, we will treat closed manifolds and manifolds with boundary. The eigenfunctions of L are often localized, as a result of disorder of the potential V, the matrix of coefficients A, irregularities of the boundary, or all of the above. In earlier work, two of us introduced the function u solving Lu=1, and showed numerically that it strongly reflects this localization. In this paper, we deepen the connection between the eigenfunctions and this landscape function u by proving that its reciprocal 1/u acts as an effective potential. The effective potential governs the exponential decay of the eigenfunctions of the system and delivers information on the distribution of eigenvalues near the bottom of the spectrum.

Keywords: Agmon distance, Schrödinger equation, spectrum, the landscape of localization

The approximation of the eigenvalues and eigenfunctions of an elliptic operator is a key computational task in many areas of applied mathematics and computational physics. An important case, especially in quantum physics, is the computation of the spectrum of a Schrödinger operator with a disordered potential. Unlike plane waves or Bloch waves that arise as Schrödinger eigenfunctions for periodic and other ordered potentials, for many forms of disordered potentials the eigenfunctions remain essentially localized in a very small subset of the initial domain. A celebrated example is Anderson localization, for which, in a continuous version, the potential is a piecewise constant function on a uniform grid whose values are sampled independently from a uniform random distribution. We present here a new method for approximating the eigenvalues and the subregions which support such localized eigenfunctions. This approach is based on the recent theoretical tools of the localization landscape and effective potential. The approach is deterministic in the sense that the approximations are calculated based on the examination of a particular realization of a random potential, and predict quantities that depend sensitively on the particular realization, rather than furnishing statistical or probabilistic results about the spectrum associated to a family of potentials with a certain distribution. These methods, which have only been partially justified theoretically, enable the calculation of the locations and shapes of the approximate supports of the eigenfunctions, the approximate values of many of the eigenvalues, and of the eigenvalue counting function and density of states, all at the cost of solving a single source problem for the same elliptic operator. We study the effectiveness and limitations of the approach through extensive computations in one and two dimensions, using a variety of piecewise constant potentials with values sampled from various different correlated or uncorrelated random distributions.

Keywords: localization, spectrum, eigenvalue, Schrödinger operator

In this paper, we consider the extension of the finite element exterior calculus from elliptic problems, in which the Hodge Laplacian is an appropriate model problem, to parabolic problems, for which we take the Hodge heat equation as our model problem. The numerical method we study is a Galerkin method based on a mixed variational formulation and using as subspaces the same spaces of finite element differential forms which are used for elliptic problems. We analyze both the semidiscrete and a fully-discrete numerical scheme.

Keywords: finite element exterior calculus, mixed finite element method, parabolic equation, Hodge heat equation

The scalar and vector Laplacians are basic operators in physics and engineering. In applications, they show up frequently perturbed by lower-order terms. The effect of such perturbations on mixed finite element methods in the scalar case is well-understood, but that in the vector case is not. In this paper, we first show that surprisingly for certain elements there is degradation of the convergence rates with certain lower-order terms even when both the solution and the data are smooth. We then give a systematic analysis of lower-order terms in mixed methods by extending the Finite Element Exterior Calculus (FEEC) framework, which contains the scalar, vector Laplacian, and many other elliptic operators as special cases. We prove that stable mixed discretization remains stable with lower-order terms for sufficiently fine discretization. Moreover, we derive sharp improved error estimates for each individual variable. In particular, this yields new results for the vector Laplacian problem which are useful in applications such as electromagnetism and acoustics modeling. Further our results imply many previous results for the scalar problem and thus unifies them all under the FEEC framework.

Keywords: finite element exterior calculus, lower order terms

The amplitude of localized quantum states in random or disordered media may exhibit long range exponential decay. We present here a theory that unveils the existence of an effective potential which finely governs the confinement of these states. In this picture, the boundaries of the localization subregions for low energy eigenfunctions correspond to the barriers of this effective potential, and the long range exponential decay characteristic of Anderson localization is explained as the consequence of multiple tunneling in the dense network of barriers created by this effective potential. Finally, we show that the Weyl’s formula based on this potential turns out to be a remarkable approximation of the density of states for a large variety of one-dimensional systems, periodic or random.

Keywords: Anderson localization, effective potential, Weyl's law, disordered media

We study the approximation properties of a wide class of finite element differential forms on curvilinear cubic meshes in n dimensions. Specifically, we consider meshes in which each element is the image of a cubical reference element under a diffeomorphism, and finite element spaces in which the shape functions and degrees of freedom are obtained from the reference element by pullback of differential forms. In the case where the diffeomorphisms from the reference element are all affine, i.e., mesh consists of parallelotopes, it is standard that the rate of convergence in L2 exceeds by one the degree of the largest full polynomial space contained in the reference space of shape functions. When the diffeomorphism is multilinear, the rate of convergence for the same space of reference shape function may degrade severely, the more so when the form degree is larger. The main result of the paper gives a sufficient condition on the reference shape functions to obtain a given rate of convergence.

Keywords: 65N30

We present stable mixed finite elements for planar linear elasticity on general quadrilateral meshes. The symmetry of the stress tensor is imposed weakly and so there are three primary variables, the stress tensor, the displacement vector field, and the scalar rotation. We develop and analyze a stable family of methods, indexed by an integer r > 1 and with rate of convergence in the L2 norm of order r for all the variables. The methods use Raviart-Thomas elements for the stress, piecewise tensor product polynomials for the displacement, and piecewise polynomials for the rotation. We also present a simple first order element, not belonging to this family. It uses the lowest order BDM elements for the stress, and piecewise constants for the displacement and rotation, and achieves first order convergence for all three variables.

Keywords: mixed finite element method, linear elasticty, quadrilateral elements

This expository article discusses the meaning of consistency, stability, and convergence of numerical discretizations of differential equations. It provides a general framework for quantifying them in which the fundamental theorem that consistency and stability imply convergence can be rigorously stated and proved. These concepts are illustrated with examples coming from both finite difference methods and finite element methods.

Keywords: consistency, stability, convergence, discretization

This expository article describes the modeling of the flight of a golf ball, with particular attention to the resolution of the drag crisis, the role of the dimples, and the optimization of the ball.

Keywords: golf, drag crisis

This article presents and explains the periodic table of finite elements.

Keywords: finite element, exterior calculus

We analyze the application to elastodynamic problems of mixed finite element methods for elasticity with weak symmetry. Our approach leads to a semidiscrete method which consists of a system of ordinary differential equations without algebraic constraints. Our error analysis, which is based on a new elliptic projection operator, applies to several mixed finite element spaces developed for elastostatics. The error estimates we obtain are robust for nearly incompressible materials.

Keywords: mixed finite element, elastodynamics, weak symmetry

We develop a family of finite element spaces of differential forms defined on cubical meshes in any number of dimensions. The family contains elements of all polynomial degrees and all form degrees. In two dimensions, these include the serendipity finite elements and the rectangular BDM elements. In three dimensions they include a recent generalization of the serendipity spaces, and new H(curl) and H(div) finite element spaces. Spaces in the family can be combined to give finite element subcomplexes of the de Rham complex which satisfy the basic hypotheses of the finite element exterior calculus, and hence can be used for stable discretization of a variety of problems. The construction and properties of the spaces are established in a uniform manner using finite element exterior calculus.

Keywords: 65N30

In 1976, Dodziuk and Patodi employed Whitney forms to define a combinatorial codifferential operator on cochains, and they raised the question whether it is consistent in the sense that for a smooth enough differential form the combinatorial codifferential of the associated cochain converges to the exterior codifferential of the form as the triangulation is refined. In 1991, Smits proved this to be the case for the combinatorial codifferential applied to 1-forms in two dimensions under the additional assumption that the initial triangulation is refined in a completely regular fashion, by dividing each triangle into four similar triangles. In this paper we extend Smits's result to arbitrary dimensions, showing that the combinatorial codifferential on 1-forms is consistent if the triangulations are uniform or piecewise uniform in a certain precise sense. We also show that this restriction on the triangulations is needed, giving a counterexample in which a different regular refinement procedure, namely Whitney's standard subdivision, is used. Further, we show by numerical example that for 2-forms in three dimensions, the combinatorial codifferential is not consistent even for the most regular subdivision process.

Keywords: consistency, combinatorial codifferential, Whitney form, finite element

This paper presents a nonconforming finite element approximation of the space of symmetric tensors with square integrable divergence, on tetrahedral meshes. Used for stress approximation together with the full space of piecewise linear vector fields for displacement, this gives a stable mixed finite element method which is shown to be linearly convergent for both the stress and displacement, and which is significantly simpler than any stable conforming mixed finite element method. The method may be viewed as the three-dimensional analogue of a previously developed element in two dimensions. As in that case, a variant of the method is proposed as well, in which the displacement approximation is reduced to piecewise rigid motions and the stress space is reduced accordingly, but the linear convergence is retained.

Keywords: mixed method; finite element; linear elasticity; nonconforming

We discuss the construction of finite element spaces of differential forms which satisfy the crucial assumptions of the finite element exterior calculus, namely that they can be assembled into subcomplexes of the de Rham complex which admit commuting projections. We present two families of spaces in the case of simplicial meshes, and two other families in the case of cubical meshes. We make use of the exterior calculus and the Koszul complex to define and understand the spaces. These tools allow us to treat a wide variety of situations, which are often treated separately, in a unified fashion.

Keywords: finite element differential form, finite element exterior calculus

We survey the reasons for the ongoing boycott of the publisher Elsevier. We examine Elsevier's pricing and bundling policies, restrictions on dissemination by authors, and lapses in ethics and peer review, and we conclude with thoughts about the future of mathematical publishing.

We consider the finite element solution of the vector Laplace equation on a domain in two dimensions. For various choices of boundary conditions, it is known that a mixed finite element method, in which the rotation of the solution is introduced as a second unknown, is advantageous, and appropriate choices of mixed finite element spaces lead to a stable, optimally convergent discretization. However, the theory that leads to these conclusions does not apply to the case of Dirichlet boundary conditions, in which both components of the solution vanish on the boundary. We show, by computational example, that indeed such mixed finite elements do not perform optimally in this case, and we analyze the suboptimal convergence that does occur. As we indicate, these results have implications for the solution of the biharmonic equation and of the Stokes equations using a mixed formulation involving the vorticity.

Keywords: vector Laplacian, Hodge Laplacian, mixed finite elements

This article discusses some of the challenges the mathematical community faces in maintaining ethical scholarly publishing, and some recent actions that have been made to confront them.

Keywords: research integrity, publishing

We investigate the journal impact factor, focusing on the applied mathematics category. We discuss impact factor manipulation and demonstrate that the impact factor gives an inaccurate view of journal quality, which is poorly correlated with expert opinion.

Keywords: impact factor

We give a new, simple, dimension-independent definition of the serendipity finite element family. The shape functions are the span of all monomials which are linear in at least s-r of the variables where s is the degree of the monomial or, equivalently, whose superlinear degree (total degree with respect to variables entering at least quadratically) is at most r. The degrees of freedom are given by moments of degree at most r-2d on each face of dimension d. We establish unisolvence and a geometric decomposition of the space.

Keywords: serendipity, finite element, unisolvence

This article, prepared in conjunction with Mathematics Awareness Month 2010 with the theme of Mathematics and Sports, reviews several examples of how mathematics eluciates physical phenomena pertaining to the golf drive. Specifically it discusses the double-pendulum model of a golf swing, transfer of energy and momentum in the club head/ball impact, and drag and lift in the flight of the golf ball.

Keywords: golf, mathematical modeling

This article reports on the confluence of two streams of research, one emanating from the fields of numerical analysis and scientific computation, the other from topology and geometry. In it we consider the numerical discretization of partial differential equations that are related to differential complexes so that de Rham cohomology and Hodge theory are key tools for exploring the well-posedness of the continuous problem. The discretization methods we consider are finite element methods, in which a variational or weak formulation of the PDE problem is approximated by restricting the trial subspace to an appropriately constructed piecewise polynomial subspace. After a brief introduction to finite element methods, we develop an abstract Hilbert space framework for analyzing the stability and convergence of such discretizations. In this framework, the differential complex is represented by a complex of Hilbert spaces and stability is obtained by transferring Hodge theoretic structures that ensure well-posedness of the continuous problem from the continuous level to the discrete. We show stable discretization discretization is achieved if the finite element spaces satisfy two hypotheses: they can be arranged into a subcomplex of this Hilbert complex, and there exists a bounded cochain projection from that complex to the subcomplex. In the next part of the paper, we consider the most canonical example of the abstract theory, in which the Hilbert complex is the de Rham complex of a domain in Euclidean space. We use the Koszul complex to construct two families of finite element differential forms, show that these can be arranged in subcomplexes of the de Rham complex in numerous ways, and for each construct a bounded cochain projection. The abstract theory therefore applies to give the stability and convergence of finite element approximations of the Hodge Laplacian. Other applications are considered as well, especially the elasticity complex and its application to the equations of elasticity. Background material is included to make the presentation self-contained for a variety of readers.

Keywords: finite element exterior calculus, exterior calculus, de Rham cohomology, Hodge theory, Hodge Laplacian, mixed finite elements

An editorial on integrity in mathematical scholarly publishing

Keywords: scholarly publishing, integrity, plagiarism, impact factor

The stability properties of simple element choices for the mixed formulation of the Laplacian are investigated numerically. The element choices studied use vector Lagrange elements, i.e., the space of continuous piecewise polynomials vector fields of degree at most r, for the vector variable, and divergence of this space, which consists of discontinuous piecewise polynomials of one degree lower, for the scalar variable. For polynomial degrees r equal 2 or 3, this pair of spaces was found to be stable for all mesh families tested. In particular, it is stable on diagonal mesh families, in contrast to its behaviour for the Stokes equations. For degree r equal 1, stability holds for some meshes, but not for others. Additionally, convergence was observed precisely for the methods that were observed to be stable. However, it seems that optimal order L2 estimates for the vector variable, known to hold for r>3, do not hold for lower degrees.

Keywords: mixed finite elements, Lagrange finite elements, stability

We study the two primary families of spaces of finite element differential forms with respect to a simplicial mesh in any number of space dimensions. These spaces are generalizations of the classical finite element spaces for vector fields, frequently referred to as Raviart-Thomas, Brezzi-Douglas-Marini, and Nédélec spaces. In the present paper, we derive geometric decompositions of these spaces which lead directly to explicit local bases for them, generalizing the Bernstein basis for ordinary Lagrange finite elements. The approach applies to both families of finite element spaces, for arbitrary polynomial degree, arbitrary order of the differential forms, and an arbitrary simplicial triangulation in any number of space dimensions. A prominent role in the construction is played by the notion of a consistent family of extension operators, which expresses in an abstract framework a sufficient condition for deriving a geometric decomposition of a finite element space leading to a local basis.

Keywords: finite element exterior calculus, finite element bases, Bernstein bases

The authors' video, also titled Moebius Transformations Revealed, presents a visualization of these complex functions. It has attracted the attention of the general public as well as a technical audience. Here, we explain the mathematics involved, as well as how the video was produced.

Keywords: Moebius transformations, complex mappings, stereographic projection, visualization

Opinion column concerning public interest in mathematics

We construct finite element subspaces of the space of symmetric tensors with square-integrable divergence on a three-dimensional domain. These spaces can be used to approximate the stress field in the classical Hellinger-Reissner mixed formulation of the elasticty equations, when standard discontinous finite element spaces are used to approximate the displacement field. These finite element spaces are defined with respect to an arbitrary simplicial triangulation of the domain, and there is one for each positive value of the polynomial degree used for the displacements. For each degree, these provide a stable finite element discretization. The construction of the spaces is closely tied to discretizations of the elasticity complex, and can be viewed as the three-dimensional analogue of the triangular element family for plane elasticity previously proposed by Arnold and Winther.

Keywords: finite element, elasticity, mixed method

A differential form is a field which assigns to each point of a domain an alternating multilinear form on its tangent space. The exterior derivative operation, which maps differential forms to differential forms of the next higher order, unifies the basic first order differential operators of calculus, and is a building block for a great variety of differential equations. When discretizing such differential equations by finite element methods, stable discretization depends on the development of spaces of finite element differential forms. As revealed recently through the finite element exterior calculus, for each order of differential form, there are two natural families of finite element subspaces associated to a simplicial triangulation. In the case of forms of order zero, which are simply functions, these two families reduce to one, which is simply the well-known family of Lagrange finite element subspaces of the first order Sobolev space. For forms of degree 1 and of degree n-1 (where n is the space dimension), we obtain two natural families of finite element subspaces, unifying many of the known mixed finite element spaces developed over the last decades.

Keywords: mixed method, finite element, differential form, exterior calculus

In this paper, we construct new finite element methods for the approximation of the equations of linear elasticity in three space dimensions that produce direct approximations to both stresses and displacements. The methods are based on a modified form of the Hellinger-Reissner variational principle that only weakly imposes the symmetry condition on the stresses. Although this approach has been previously used by a number of authors, a key new ingredient here is a constructive derivation of the elasticity complex starting from the de Rham complex. By mimicking this construction in the discrete case, we derive new mixed finite elements for elasticity in a systematic manner from known discretizations of the de Rham complex. These elements appear to be simpler than the ones previously derived. For example, we construct stable discretizations which use only piecewise linear elements to approximate the stress field and piecewise constant functions to approximate the displacement field.

Keywords: mixed method, finite element, elasticity

Specifying boundary conditions continues to be a challenge in numerical relativity in order to obtain a long time convergent numerical simulation of Einstein's equations in domains with artificial boundaries. In this paper, we address this problem for the Einstein-Christoffel (EC) symmetric hyperbolic formulation of Einstein's equations linearized around flat spacetime. First, we prescribe simple boundary conditions that make the problem well posed and preserve the constraints. Next, we indicate boundary conditions for a system that extends the linearized EC system by including the momentum constraints and whose solution solves Einstein's equations in a bounded domain. Finally, we extend our results to the case of inhomogeneous boundary conditions.

Keywords: general relativity, Einstein equations, boundary condition

In a recent paper of Arnold, Brezzi, and Marini, the ideas of discontinuous Galerkin methods were used to obtain and analyze two new families of locking free finite element methods for the approximation of the Reissner-Mindlin plate problem. By following their basic approach, but making different choices of finite element spaces, we develop and analyze other families of locking free finite elements that eliminate the need for the introduction of a reduction operator, which has been a central feature of many locking-free methods. For k>1, all the methods use piecewise polynomials of degree k to approximate the transverse displacement and (possibly subsets) of piecewise polynomials of degree k-1 to approximate both the rotation and shear stress vectors. The approximation spaces for the rotation and the shear stress are always identical. The methods vary in the amount of interelement continuity required. In terms of smallest number of degrees of freedom, the simplest method approximates the transverse displacement with continuous, piecewise quadratics and both the rotation and shear stress with rotated linear Brezzi-Douglas-Marini elements.

Keywords: discontinuous Galerkin, Reissner-Mindlin plate, locking

Finite element exterior calculus is an approach to the design and understanding of finite element discretizations for a wide variety of systems of partial differential equations. This approach brings to bear tools from differential geometry, algebraic topology, and homological algebra to develop discretizations which are compatible with the geometric, topological, and algebraic structures which underlie well-posedness of the PDE problem being solved. In the finite element exterior calculus, many finite element spaces are revealed as spaces of piecewise polynomial differential forms. These connect to each other in discrete subcomplexes of elliptic differential complexes, and are also related to the continuous elliptic complex through projections which commute with the complex differential. Applications are made to the finite element discretization of a variety of problems, including the Hodge Laplacian, Maxwell's equations, the equations of elasticity, and elliptic eigenvalue problems, and also to preconditioners.

Keywords: finite element, exterior calculus, mixed method, Hodge, de Rham

A close connection between the ordinary de Rham complex and a corresponding elasticity complex is utilized to derive new mixed finite element methods for linear elasticity. For a formulation with weakly imposed symmetry, this approach leads to methods which are simpler than those previously obtained. For example, we construct stable discretizations which use only piecewise linear elements to approximate the stress field and piecewise constant functions to approximate the displacement field. We also discuss how the strongly symmetric methods proposed in [8] can be derived in the present framework. The method of construction works in both two and three space dimensions, but for simplicity the discussion here is limited to the two dimensional case.

Keywords: mixed finite element method, Hellinger-Reissner principle, elasticity

In this paper we explain the relation between certain piecewise polynomial subcomplexes of the de Rham complex and the stability of mixed finite element methods for elliptic problems.

Keywords: mixed finite element method, de Rham complex, stability

We present a family of stable rectangular mixed finite elements for plane elasticity. Each member of the family consists of a space of piecewise polynomials discretizing the space of symmetric tensors in which the stress field is sought, and another to discretize the space of vector fields in which the displacement is sought. These may be viewed as analogues in the case of rectangular meshes of mixed finite elements recently proposed for triangular meshes. As for the triangular case the elements are closely related to a discrete version of the elasticity differential complex.

Keywords: mixed method, finite element, elasticity, rectangular

We develop a family of locking-free elements for the Reissner-Mindlin plate using Discontinuous Galerkin techniques, one for each odd degree, and prove optimal error estimates. A second family uses conforming elements for the rotations and nonconforming elements for the transverse displacement, generalizing the element of Arnold and Falk to higher degree.

Keywords: discontinuous Galerkin, Reissner-Mindlin plate, locking

We consider the approximation properties of quadrilateral finite element spaces of vector fields defined by the Piola transform, extending results previously obtained for scalar approximation. The finite element spaces are constructed starting with a given finite dimensional space of vector fields on a square reference element, which is then transformed to a space of vector fields on each convex quadrilateral element via the Piola transform associated to a bilinear isomorphism of the square onto the element. For affine isomorphisms, a necessary and sufficientcondition for approximation of order r+1 in L2 is that each component ofthe given space of functions on the reference element contain all polynomial functions of total degree at most r. In the case of bilinear isomorphisms,the situation is more complicated and we give a precise characterization of what is needed for optimal order L2-approximation of the function and of its divergence. As applications, we demonstrate degradation of the convergence order on quadrilateral meshes as compared to rectangular meshes for some standard finite element approximations of H(div). We also derive new estimates for approximation by quadrilateral Raviart-Thomas elements (requiring less regularity) and propose a new quadrilateral finite element space which provides optimal order approximation in H(div). Finally, we demonstrate the theory with numerical computations of mixed and least squares finite element aproximations of the solution of Poisson's equation.

Keywords: quadrilateral, finite element, approximation, mixed finite element

We derive a new first-order formulation for Einstein's equations which involves fewer unknowns than other first-order formulations that have been proposed. The new formulation is based on the 3+1 decomposition with arbitrary lapse and shift. In the reduction to first order form only 8 particular combinations of the 18 first derivatives of the spatial metric are introduced. In the case of linearization about Minkowski space, the new formulation consists of symmetric hyperbolic system in 14 unknowns, namely the components of the extrinsic curvature perturbation and the 8 new variables, from whose solution the metric perturbation can be computed by integration.

Keywords: relativity, Einstein equations, symmetric hyperbolic

In this paper we propose a way to analyze certain classes of dimension reduction models for elliptic problems in thin domains. We develop asymptotic expansions for the exact and model solutions, having the thickness as small parameter. The modeling error is then estimated by comparing the respective expansions, and the upper bounds obtained make clear the influence of the order of the model and the thickness on the convergence rates. The techniques developed here allows for estimates in several norms and semi-norms, and also interior estimates (which disregards boundary layers).

Keywords: hierarchical modeling, dimension reduction, asymptotic estimates

We present a family of pairs of finite element spaces for the unaltered Hellinger-Reissner variational principle using polynomial shape functions on a single triangular mesh for stress and displacement. There is a member of the family for each polynomial degree, beginning with degree two for the stress and degree one for the displacement, and each is stable and affords optimal order approximation. The simplest element pair involves 24 local degrees of freedom for the stress and 6 for the displacement. We also construct a lower order element involving 21 stress degrees of freedom and 3 displacement degrees of freedom which is, we believe, likely to be the simplest possible conforming stable element pair with polynomial shape functions. For all these conforming elements the approximate stress not only belongs to H(div), but is also continuous at element vertices, which is more continuity than may be desired. We show that for conforming finite elements with polynomial shape functions, this additional continuity is unavoidable. To overcome this obstruction, we construct as well some non-conforming stable mixed finite elements, which we show converge with optimal order as well. The simplest of these involves only 12 stress and 6 displacement degrees of freedom on each triangle.

Keywords: mixed method, finite element, elasticity

We construct first order, stable, nonconforming mixed finite elements for plane elasticity and analyze their convergence. The mixed method is based on the Hellinger-Reissner variational formulation in which the stress and displacement fields are the primary unknowns. The stress elements use polynomial shape functions but do not involve vertex degrees of freedom.

Keywords: mixed method, finite element, nonconforming, elasticity

Differential complexes such as the de Rham complex have recently come to play an important role in the design and analysis of numerical methods for partial differential equations. The design of stable discretizations of systems of partial differential equations often hinges on capturing subtle aspects of the structure of the system in the discretization. In many cases the differential geometric structure captured by a differential complex has proven to be a key element, and a discrete differential complex which is appropriately related to the original complex is essential. This new geometric viewpoint has provided a unifying understanding of a variety of innovative numerical methods developed over recent decades and pointed the way to stable discretizations of problems for which none were previously known, and it appears likely to play an important role in attacking some currently intractable problems in numerical PDE.

Keywords: finite element, numerical stability, differential complex

Over the last two decades, there has been an extensive effort to devise and analyze finite elements schemes for the approximation of the Reissner�Mindlin plate equations which avoid locking, numerical overstiffness resulting in a loss of accuracy when the plate is thin. There are now many triangular and rectangular finite elements, for which a mathematical analysis exists to certify them as free of locking. Generally speaking, the analysis for rectangular elements extends to the case of parallograms, which are defined by affine mappings of rectangles. However, for more general convex quadrilaterals, defined by bilinear mappings of rectangles, the analysis is more complicated. Recent results by the authors on the approximation properties of quadrilateral finite elements shed some light on the problems encountered. In particular, they show that for some finite element methods for the approximation of the Reissner-Mindlin plate, the obvious generalization of rectangular elements to general quadrilateral meshes produce methods which lose accuracy. In this paper, we present an overview of this situation.

Keywords: Reissner-Mindlin plate, finite element, locking-free, isoparametric

We show that the Reissner-Mindlin plate bending model has a wider range of applicability than the Kirchhoff-Love model for the approximation of clamped linearly elastic plates. Under the assumption that the body force density is constant in the transverse direction, the Reissner-Mindlin model solution converges to the three-dimensional linear elasticity solution in the relative energy norm for the full range of surface loads. However, for loads with a significant transverse shear effect, the Kirchhoff-Love model fails.

Keywords: plate, Reissner-Mindlin, Kirchhoff-Love

We provide a framework for the analysis of a large class of discontinuous methods for second-order elliptic problems. It allows for the understanding and comparison of most of the discontinuous Galerkin methods that have been proposed for the numerical treatment of elliptic problems by diverse communities over three decades.

Keywords: elliptic problems, discontinuous Galerkin, interior penalty

There have been many efforts, dating back four decades, to develop stable mixed finite elements for the stress-displacement formulation of the plane elasticity system. This requires the development of a compatible pair of finite element spaces, one to discretize the space of symmetric tensors in which the stress field is sought, and one to discretize the space of vector fields in which the displacement is sought. Although there are number of well-known mixed finite element pairs known for the analogous problem involving vector fields and scalar fields, the symmetry of the stress field is a substantial additional difficulty, and the elements presented here are the first ones using polynomial shape functions which are known to be stable. We present a family of such pairs of finite element spaces, one for each polynomial degree, beginning with degree two for the stress and degree one for the displacement, and show stability and optimal order approximation. We also analyze some obstructions to the construction of such finite element spaces, which account for the paucity of elements available.

Keywords: mixed method, finite element, elasticity

We consider the approximation properties of finite element spaces on quadrilateral meshes. The finite element spaces are constructed starting with a given finite dimensional space of functions on a square reference element, which is then transformed to a space of functions on each convex quadrilateral element via a bilinear isomorphism of the square onto the element. It is known that for affine isomorphisms, a necessary and sufficient condition for approximation of order r+1 in L2 and order r in H1 is that the given space of functions on the reference element contain all polynomial functions of total degree at most r. In the case of bilinear isomorphisms, it is known that the same estimates hold if the function space contains all polynomial functions of separate degree r. We show, by means of a counterexample, that this latter condition is also necessary. As applications we demonstrate degradation of the convergence order on quadrilateral meshes as compared to rectangular meshes for serendipity finite elements and for various mixed and nonconforming finite elements.

Keywords: quadrilateral, finite element, approximation, serendipity, mixed finite element

Quadrilateral finite elements are generally constructed by starting from a given finite dimensional space of polynomials V^ on the unit reference square K^. The elements of V^ are then transformed by using the bilinear isomorphisms F_K which map K^ to each convex quadrilateral element K. It has been recently proven that a necessary and sufficient condition for approximation of order r+1 in L^2 and r in H^1 is that V^ contains the space Q_r of all polynomial functions of degree r separately in each variable. In this paper several numerical experiments are presented which confirm the theory. The tests are taken from various examples of applications: the Laplace operator, the Stokes problem and an eigenvalue problem arising in fluid-structure interaction modeling.

Keywords: quadrilateral, finite element, approximation, serendipity, mixed finite element

The construction of gravitational wave observatories is one of the greatest scientific efforts of our time. As a result, there is presently a strong need to numerically simulate the emission of gravitation radiation from massive astronomical events such as black hole collisions. This entails the numerical solution of the Einstein field equations. We briefly describe the field equations in their natural setting, namely as statements about the geometry of space time. Next we describe the complicated system that arises when the field equations are recast as partial differential equations, and discuss procedures for deriving from them a more tractable system consisting of constraint equations to be satisfied by initial data and together with evolution equations. We present some applications of modern finite element technology to the solution of the constraint equations in order to find initial data relevant to black hole collisions. We conclude by enumerating some of the many computational challenges that remain.

Keywords: general relativity, gravitational radiation, black holes, Einstein equations

We provide a common framework for the understanding, comparison, and analysis of several discontinuous Galerkin methods that have been proposed for the numerical treatment of elliptic problems. This class includes the recently introduced methods of Bassi and Rebay (together with the variants proposed by Brezzi, Manzini, Marini, Pietra and Russo), the local discontinuous Galerkin methods of Cockburn and Shu, and the method of Baumann and Oden. It also includes the so-called interior penalty methods developed some time ago by Douglas and Dupont, Wheeler, Baker, and Arnold among others.

We consider the solution of systems of linear algebraic equations which arise from the finite element discretization of variational problems posed in the Hilbert spaces H(div) and H(curl) in three dimensions. We show that if appropriate finite element spaces and appropriate additive or multiplicative Schwarz smoothers are used, then the multigrid V-cycle is an efficient solver and preconditioner for the discrete operator. All results are uniform with respect to the mesh size, the number of mesh levels, and weights on the two terms in the inner products.

Keywords: multigrid, preconditioner, mixed method, finite element

We present an algorithm for the construction of locally adapted conformal tetrahedral meshes. The algorithm is based on bisection of tetrahedra. A new data structure is introduced, which simplifies both the selection of the refinement edge of a tetrahedron and the recursive refinement to conformity of a mesh once some tetrahedra have been bisected. We prove that repeated application of the algorithm leads to only finitely many tetrahedral shapes up to similarity, and bound the amount of additional refinement that is needed to achieve conformity. Numerical examples of the effectiveness of the algorithm are presented.

Keywords: bisection, tetrahedral meshes, adaptive refinement, similarity classes, finite elements

An adaptive finite element algorithm for elliptic boundary value problems in ³ is presented. The algorithm uses linear finite elements, a-posteriori error estimators, a mesh refinement scheme based on bisection of tetrahedra, and a multi-grid solver. We show that the repeated bisection of an arbitrary tetrahedron leads to only a finite number of dissimilar tetrahedra, and that the recursive algorithm ensuring conformity of the meshes produced terminates in a finite number of steps. A procedure for assigning numbers to tetrahedra in a mesh based on a-posteriori error estimates, indicating the degree of refinement of the tetrahedron, is also presented. Numerical examples illustrating the effectiveness of the algorithm are given.

Keywords: finite elements, adaptive mesh refinement, error estimators, bisection of tetrahedra

According to the theory of general relativity, the relative acceleration of masses generates gravitational radiation. Although gravitational radiation has not yet been detected, it is believed that extremely violent cosmic events, such as the collision of black holes, should generate gravity waves of sufficient amplitude to detect on earth. The massive Laser Interferometer Gravitational-Wave Observatory, or LIGO, is now being constructed to detect gravity waves. Consequently there is great interest in the computer simulation of black hole collisions and similar events, based on the numerical solution of the Einstein field equations. In this note we introduce the scientific, mathematical, and computational problems and discuss the development of a computer code to solve the initial data problem for colliding black holes, a nonlinear elliptic boundary value problem posed in an unbounded three dimensional domain which is a key step in solving the full field equations. The code is based on finite elements, adaptive meshes, and a multigrid solution process. Here we will particularly emphasize the mathematical and algorithmic issues arising in the generation of adaptive tetrahedral meshes.

Keywords: adaptivity, finite elements, black holes, Einstein equations

In an earlier paper we constructed and analyzed a multigrid preconditioner for the system of linear algebraic equations arising from the finite element discretization of boundary value problems associated to the differential operator I - grad div. In this paper we analyze the procedure without assuming that the underlying domain is convex and show that, also in this case, the preconditioner is spectrally equivalent to the inverse of the discrete operator.

Keywords: preconditioner, finite element, multigrid, nonconvex domain

We consider the derivation and rigorous justification of models for thin linearly elastic plates using mixed variational principles.

Keywords: plate, dimensional reduction, mixed variational principle

We consider iterative methods for the solution of linear systems of equations arising from mixed finite element discretization of the Reissner-Mindlin plate model. We show how to construct symmetric positive definite block diagonal preconditioners for these indefinite systems such that the resulting systems have spectral condition numbers independent of both the mesh size h and the plate thickness t.

Keywords: preconditioner, Reissner, Mindlin, plate, finite element

Summarizing the work of [1], we show how to construct preconditioners using domain decomposition and multigrid techniques for the system of linear algebraic equations which arises from the finite element discretization of boundary value problems associated to the differential operator I - grad div. These preconditioners are shown to be spectrally equivalent to the inverse of the operator and thus may be used to precondition iterative methods so that any given error reduction may be achieved in a finite number of iterations independent of the mesh discretization. We describe applications of these results to the efficient solution of mixed and least squares finite element approximations of elliptic boundary value problems.

Keywords: preconditioner, mixed method, least squares, finite element,

We consider the derivation of two-dimensional models for the bending and stretching of a thin three-dimensional linearly elastic plate using variational methods. Specifically we consider restriction of the trial space in two different forms of the Hellinger-Reissner variational principle for 3-D elasticity to functions with a specified polynomial dependence in the transverse direction. Using this approach many different plate models are possible and we classify and investigate the most important. We study in detail a method which leads naturally not only to familiar plate models, but also to error bounds between the plate solution and the full 3-D solution.

Keywords: plate, dimensional reduction, Reissner-Mindlin

Encyclopedia article on computer-aided instruction

We consider iterative methods for the solution of the linear system of equations arising from the mixed finite element discretization of the Reissner-Mindlin plate model. We show how to construct a symmetric positive definite block diagonal preconditioner such that the resulting linear system has spectral condition number independent of both the mesh size h and the plate thickness t. We further discuss how this preconditioner may be implemented and then apply it to efficiently solve this indefinite linear system. Although the mixed formulation of the Reissner-Mindlin problem has a saddle-point structure common to other mixed variational problems, the presence of the small parameter t and the fact that the matrix in the upper left corner of the partition is only positive semidefinite introduces new complications.

Keywords: preconditioner, Reissner, Mindlin, plate, finite element

We consider the solution of the system of linear algebraic equations which arises from the finite element discretization of boundary value problems associated to the differential operator I - grad div. The natural setting for such problems is in the Hilbert space H(div) and the variational formulation is based on the inner product in H(div). We show how to construct preconditioners for these equations using both domain decomposition and multigrid techniques. These preconditioners are shown to be spectrally equivalent to the inverse of the operator. As a consequence, they may be used to precondition iterative methods so that any given error reduction may be achieved in a finite number of iterations, with the number independent of the mesh discretization. We describe applications of these results to the efficient solution of mixed and least squares finite element approximations of elliptic boundary value problems.

Keywords: preconditioner, mixed method, least squares, finite element, multigrid, domain decomposition

An analysis is presented for a recently proposed finite element method for the Reissner-Mindlin plate problem. The method is based on the standard variational principle, uses nonconforming linear elements to approximate the rotations and conforming linear elements to approximate the transverse displacements, and avoids the usual "locking problem" by interpolating the shear stress into a rotated space of lowest order Raviart-Thomas elements. When the plate thickness t=O(h), it is proved that the method gives optimal order error estimates uniform in t. However, the analysis suggests and numerical calculations confirm that the method can produce poor approximations for moderate sized values of the plate thickness. Indeed, for t fixed, the method does not converge as the mesh size h tends to zero.

Keywords: Reissner, Mindlin, plate, finite element, nonconforming

Interior error estimates are obtained for a low order finite element introduced by Arnold and Falk for the Reissner-Mindlin plates. It is proved that the approximation error of the finite element solution in the interior domain is bounded above by two parts: one measures the local approximability of the exact solution by the finite element space and the other the global approximability of the finite element method. As an application, we show that for the soft simply supported plate, the Arnold-Falk element still achieves an almost optimal convergence rate in the energy norm away from the boundary layer, even though optimal order convergence cannot hold globally due to the boundary layer. Numerical results are given which support our conclusion.

Keywords: Reissner-Mindlin plate, boundary layer, mixed finite element,

We briefly present the main idea of partial selective reduced integration as developed in other works of the authors. The idea is quite general and can be applied to a number of different situations, but we concentrate on the case of the Naghdi shell model.

Keywords: partial selective reduced integration, Naghdi shell

We propose a new family of finite element methods for the Naghdi shell model, one method associated with each nonnegative integer k. The methods are based on a nonstandard mixed formulation, and the kth method employs triangular Lagrange finite elements of degree k+2 augmented by bubble functions of degree k+3 for both the displacement and rotation variables, and discontinuous piecewise polynomials of degree k for the shear and membrane stresses. This method can be implemented in terms of the displacement and rotation variables alone, as the minimization of an altered energy functional over the space mentioned. The alteration consists of the introduction of a weighted local projection into part, but not all, of the shear and membrane energy terms of the usual Naghdi energy. The relative error in the method, measured in a norm which combines the H1 norm of the displacement and rotation fields and an appropriate norm of the shear and membrane stress fields, converges to zero with order k+1 uniformly with respect to the shell thickness for smooth solutions, at least under the assumption that certain geometrical coefficients in the Nagdhi model are replaced by piecewise constants.

Keywords: shell, locking, finite element

We investigate the structure of the solution of the Reissner-Mindlin plate equations in its dependence on the plate thickness for various boundary conditions, developing asymptotic expansions in powers of the plate thickness for the main physical quantities. These expansions are uniform up to the boundary for the transverse displacement, but for other variables there is a boundary layer, whose strength depends on the boundary conditions. We give rigorous error bounds for the errors in the expansions in Sobolev norms and make various applications.

Keywords: Reissner, Mindlin, plate, boundary layer

Local error estimates are derived which apply to most stable finite mixed finite element discretizations of the stationary Stokes equations.

Keywords: Stokes equations, mixed finite element method, local

Nonconforming piecewise linear finite elements for the velocity field and piecewise constant elements for the pressure field give a simple stable, optimal order approximation to the Stokes equations, but are not stable for the equations of incompressible elasticty, which differ from the Stokes equations only in that the vector Laplace operator is replaced by the Lame operator. However, we show that if we replace the divergence by the rotation, then the nonconforming linear-constant element is stable both for the system involving the Laplacian and for that involving the Lame operator. Finally we discuss an application to the Reissner-Mindlin plate.

Keywords: Stokes equations, mixed finite element

We report on a new approach to obtaining stable locking free finite element discretizations for the Reissner-Mindlin plate.

Keywords: Reissner-Mindlin plate, finite element

We study the finite element approximation of the stationary Stokes equations in the velocity-pressure formulation using continuous piecewise quadratic functions for velocity and discontinuous piecewise linear functions for pressure. For some meshes this method is unstable, even after spurious pressure modes are removed. For other meshes there are spurious local pressure modes, but once they are removed the method is stable, and in particular, the velocity converges with optimal order. On yet other meshes there are no spurious pressure modes and the method is stable and optimally convergent for both pressure and velocity.

Keywords: finite element, Stokes equations

Finite element methods for the Reissner-Mindlin plate theory are discussed. Methods in which both the tranverse displacement and the rotation are approximated by finite elements of low degree mostly suffer from locking. However a number of related methods have been devised recently which avoid locking effects. Although the finite element spaces for both the rotation and transverse displacement contain little more than piecewise linear functions, optimal order convergence holds uniformly in the thickness. The main ideas leading to such methods are reviewed and the relationships between various methods are clarified.

Keywords: Reissner, Mindlin, plate, finite element

This paper treats the basic ideas of mixed finite element methods at an introductory level. Although the viewpoint presented is that of a mathematician, the paper is aimed at practitioners and the mathematical prerequisites are kept to a minimum. A classification of variational principles and of the corresponding weak formulations and Galerkin methods-displacement, equilibrium, and mixed-is given and illustrated through four significant examples. The advantages and disadvantages of mixed methods are discussed. The concepts of convergence, approximability, and stability and their interrelations are developed, and a resume is given of the stability theory which governs the performance of mixed methods. The paper concludes with a survey of techniques that have been developed for the construction of stable mixed methods and numerous examples of such methods.

Keywords: mixed method, finite element, variational principle

The structure of the solution of the Reissner-Mindlin model of a clamped plate is investigated, emphasizing its dependence on the plate thickness. Asymptotic expansions in powers of the plate thickness are developed for the main physical quantities and the boundary layer is studied. Rigorous error bounds are given for the errors in the expansions in Sobolev norms. As applications, new regularity results for the solutions and new estimates for the difference between the Reissner-Mindlin solution and the solution to the biharmonic equation are derived. Boundary conditions for a clamped edge are considered for most of the paper, and the very similar case of a hard simply-supported plate is discussed briefly at the end.

Keywords: Reissner, Mindlin, plate, boundary layer

We investigate the structure of the solution of the Reissner-Mindlin plate equations in its dependence on the plate thickness for various boundary conditions, developing asymptotic expansions in powers of the plate thickness for the main physical quantities. These expansions are uniform up to the boundary for the transverse displacement, but for other variables there is a boundary layer, whose strength depends on the boundary conditions. We give rigorous error bounds for the errors in the expansions in Sobolev norms and make various applications.

Keywords: Reissner, Mindlin, plate, boundary layer, edge effect

The single layer heat potential operator, K, arises in the solution of initial-boundary value problems for the heat equation using boundary integral methods. In this note we show that K maps a certain anisotropic Sobolev space isomorphically onto its dual, and, moreover, satisfies the coercivity inequality <K q,q> >=c|q|^2. We thereby establish the well-posedness of the operator equation K q=f and provide a basis for the analysis of the discretizations.

Keywords: coercivity, heat potential, anisotropic Sobolev space

We present and analyze a simple finite element method for the Mindlin-Reissner plate model in the primitive variables. Our method uses nonconforming linear finite elements for the transverse displacement and conforming linear finite elements enriched by bubbles for the rotation, with the computation of the element stiffness matrix modified by the inclusion of a simple elementwise averaging. We prove that the method converges with optimal order uniformly with respect to thickness.

Keywords: Reissner-Mindlin plate, finite element, nonconforming

The Dirichlet problem for Laplace's equation is often solved by means of the single layer potential representation, leading to a Fredholm integral equation of the first kind with logarithmic kernel. We propose to solve this integral equation using a Petrov-Galerkin method with trigonometric polynomials as test functions and, as trial functions, a span of delta distributions centered at boundary points. The approximate solution to the boundary value problem thus computed converges exponentially away from the boundary and algebraically up to the boundary. We show that these convergence results hold even when the discretization matrices are computed via numerical quadratures. Finally, we discuss our implementation of this method using the fast Fourier transform to compute the discretization matrices, and present numerical experiments in order to confirm our theory and to examine the behavior of the method in cases where the theory doesn't apply due to lack of smoothness.

We consider the existence of regular solutions to the boundary value problem div U = f on a plane polygonal domain with the Dirichlet boundary condition U=g. We formulate simultaneously necessary and sufficient conditions on f and g in order that a solution U exist in the Sobolev space W^s+1_p. In addition to the obvious regularity and integral conditions these consist of at most one compatibility condition at each vertex of the polygon. In the special case of homogeneous boundary data, it is necessary and sufficient that f belong to W^s_p, have mean value zero, and vanish at each vertex. (The latter condition only applies if s is large enough that the point values make sense.) We construct a solution operator which is independent of s and p. As intermediate results we obtain various new trace theorems for Sobolev spaces on polygons.

Keywords: divergence, trace, Sobolev space

We propose a new mixed variational formulation for the equations of linear elasticity. It does not require symmetric tensors and consequently is easy to discretize by adapting mixed finite elements developed for scalar second order elliptic equations.

Keywords: mixed finite element method, elasticity

The solution of the heat equation with Dirichlet boundary conditions by the boundary integral method leads to an integral equation of the first kind to determine the boundary flux. We show that the linear operator so defined is an automorphism from a certain function to another defined on the boundary and is coercive, thereby establishing the well-posedness of the method.

Keywords: boundary integral method, heat equation

We consider the equations of linear homogeneous anisotropic elasticity admitting the possibility that the material is internally constrained, and formulate a simple necessary and sufficient condition for the fundamental boundary value problems to be well-posed. For materials fulfilling the condition, we establish continuous dependence of the displacement and stress on the elastic moduli and ellipticity of the elasticity system. As an application we determine the orthotropic materials for which the fundamental problems are well-posed in terms of their Young's moduli, shear moduli, and Poisson ratios. Finally, we derive a reformulation of the elasticity system that is valid for both constrained and unconstrained materials and involves only one scalar unknown in addition to the displacements. For a two-dimensional constrained material a further reduction to a single scalar equation is outlined.

Keywords: elasticity, anisotropic, constraint, well-posed

We prove apriori estimates and continuous dependence on the elastic moduli for the equations of homogeneous orthotropic elasticity. These results are uniform with respect to the three Poisson rations, Young's moduli, and shear moduli of the material for certain ranges of these constants. These ranges include the possibility that the compliance tensor is singular such as occurs for incompressible materials.

Keywords: orthotropic elasticity, incompressible, constrained

Most boundary element methods for two-dimensional boundary value problems are based on point collocation on the boundary and the use of splines as trial functions. Here we present a unified asymptotic error analysis for even as well as for odd degree splines subordinate to uniform or smoothly graded meshes and prove asymptotic convergence of optimal order. The equations are collocated at the breakpoints for odd degree and the internodal midpoints for even degree splines. The crucial assumption for the generalized boundary integral and integro-differential operators is strong ellipticity. Our analysis is based on simple Fourier expansions. In particular, we extend results by J. Saranen and W.L. Wendland from constant to variable coefficient equations. Our results include the first convergence proof of midpoint collocation with piecewise constant functions, i.e., the panel method for solving systems of Cauchy singular integral equations.

We discuss a technique of implementing certain mixed finite elements based on the use of Lagrange multipliers to impose interelement continuity. The matrices arising from this implementation are positive definite. Considering some well-known mixed methods, namely the Raviart-Thomas methods for second order elliptic problems and the Hellan-Hermann-Johnson method for biharmonic problems, we show that the computed Lagrange multipliers may be exploited in a simple postprocess to produce better approximation of the original variables. We further extablish an equivalence between the mixed methods and certain modified versions of well-known nonconforming methods, notably the Morley method in the case of the biharmonic problem. The equivalence is exploited to provide error estimates for both the mixed and nonconforming methods.

Keywords: mixed finite element, Lagrange multiplier

We present in this paper a new velocity-pressure finite element for the computation of Stokes flow. We discretize the velocity field iwth continuous piecewise linear functions enriched by bubble functions, and the pressure by piecewise linear functions. We show that this element satisfies the usual inf-sup condition and converges with first order for both velocities and pressures. Finally we relate this element to families of higher order elements and to the popular Taylor-Hood element.

Keywords: finite element, Stokes equations

A mixed formulation for boundary value problems in linear elastostatics is presented. This formulation differs slightly from the classical Hellinger-Reissner formulation. The unknown fields are the displacement and a tensor related but not equal to the stress. The tensors appearing in the formulation need not be symmetric, and consequently mixed finite elements developed for scalar second order elliptic problems may be applied directly.

Keywords: elasticity, mixed finite element

We investigate the asymptotic convergence properties of a variety of methods for the numerical solution of the system of singular integral equations arising from the traction problem of plane elasticity. Various sorts of Galerkin methods and collocation methods are considered, all of which determine a spline approximation via paring with certain test functions; the test functions may be splines of the same degree as the trial functions (ordinary Galerkin methods), splines of different degree (Petrov-Galerkin methods), delta functions (collocation), or trigonometric polynomials (spline-trig methods). The choice of test functions is shown to have a significant influence on the convergence properties.

Keywords: integral equations, Galerkin methods, collocation

A mixed finite element procedure for plane elasticity is introduced and analyzed. The symmetry of the stress tensor is enforced through the introduction of a Lagrange multiplier. An additional Lagrange multiplier is instroduced to simplify the algebraic system. Applications are made to incompressible elastic problems and to plasticity problems.

Keywords: finite element methods, plane elasticity

The Dirichler problem for the equations of plane elasticity is approximated by a mixed finite element method using a new family of composite finite elements having properties analogous to those possessed by the Raviart-Thomas mixed finite elements for a scalar, second-order elliptic equation. Estimates of optimal order and minimal regularity are derived for the errors in the displacement vector and the stress tensor in L^2 and optimal order negative norm estimates are obtained in (H^s)' for a range of s depending on the index of the finite element space. An optimal order estimate inL in L_infinity for the displacement error is given. Also, a quasioptimal estimate is derived in an appropriate space. All estimates are valid uniformly with respect to the compressibility and apply in the incompressible case. The formulation of the elements is presented in detail.

Keywords: finite element methods, plane elasticity

We examine the asymptotic accuracy of the method of collocation for the approximate solution of linear elliptic partial differential equations. Specifically we consider the nodal collocation of a second order equation in the plane with biperiodicity conditions using tensor product smooth splines of odd degree as trial functions. We prove optimal rates of convergence in L2 for partial derivatives of the approximate solution which are of order at least two in one variable, while the solution itself and its gradient converge in L2 at rates less than the optimal approximation theoretic results.

Principles for the selection of a finite element method for a particular problem are discusses. These principles are stated in terms of the notion of approximability, optimality, and stability. Several examples are discussed in details as illustrations. Conclusions regarding the selection of finite element methods are summarized in the final section of the paper.

Keywords: finite elements, aproximability, stability, optimality

We prove quasioptimal and optimal order estimates in various Sobolev norms for the approximation of linear strongly elliptic pseudodifferential equations in one independent variable by the method of nodal collocation by odd degree polynomial splines. The analysis pertains in particular to many of the boundary element methods used for numerical computation in engineering applications. Equations to which the analysis is applied include Fredholm integral equations of the second kind, certain first kind Fredholm equations, singular integral equations involving Cauchy kernels, a variety of integro-differential equations, and two-point boundary value problems for ordinary differential equations. The error analysis is based on an equivalence which we establish between the collocation methods and certain nonstandard Galerkin methods. We compare the collocation method with a standard Galerkin method using splines of the same degree, showing that the Galerkin method is quasioptimal in a Sobolev space of lower index and furnishes optimal order approximation for a range of Sobolev indices containing and extending below that for the collocation method, and so the standard Galerkin method achieves higher rates of convergence.

Keywords: collocation, spline, integral equation

We consider a Galerkin method for functional equations in one space variable which uses periodic cardinal splines as trial functions and trigonometric polynomials as test functions. We analyze the method applied to the integral equation of the first kind arising from a single layer potential formulation of the Dirichlet problem in the interior or exterior of an analytic plane curve. In contrast to ordinary spline Galerkin methods, we show that the method is stable, and so provides quasioptimal approximation, in a large family of Hilbert spaces including all the Sobolev spaces of negative order. As a consequence we prove that the approximate solution to the Dirichlet problem and all its derivatives converge pointwise with exponential rate.

Keywords: spline, spline-trigonometric, Galerkin method, boundary integral

The goal of engineering computations is to obtain quantitative information about engineering problems. This goal is usually achieved by the approximation solution of a mathematically formulated problem. Although a relevant mathematical formulation of the problem and its approximation solution are closely related, here we shall suppose that a mathematical formulation has already been determined and is amenable to an approximate treatment. We shall discuss a broad class of approaches based on variational methods of discretization which allow one to find the approximation solution within a desired range of accuracy. We discuss properties of these methods which enable us to distinguish among them and which aid in the selection or design of a method which is effective in achieving the given goals of the computation.

Keywords: finite elements, variational methods

We compare the efficiency of the solution of two-dimensional elliptic boundary value problems via boundary integral methods using two different discretization procedures with comparable convergence rates: Galerkin procedures with numerical integration and collocation.

Keywords: boundary integral methods, boundary element methods,

A new semidiscrete finite element method for the solution of second order nonlinear parabolic boundary value problems is formulated and analyzed. The test and trial spaces consist of discontinuous piecewise polynomial functions over quite general meshes with interelement continuity enforced approximately by means of penalties. Optimal order error estimates in energy and L2-norms are stated in terms of locally expressed quantities. They are proved first for a model problem and then in general.

An unconditionally stable fully discrete finite element method for the Korteweg-de Vries equation is presented. In addition to satisfying optimal order global estimates, it is shown that this method is superconvergent at the nodes. The algorithm is derived from the conservative method proposed by the second author by the introduction of a small time-independent forcing term into the discrete equations. This term is a form of the quasiprojection which was first employed in the analysis of superconvergence phenomena for parabolic problems. However, in the present work, unlike in the parabolic case, the quasiprojection is used as perturbation of the discrete equations and does not affect the choice of initial values.

Keywords: superconvergence, finite element, Korteweg-de Vries equation

The discretization by finite elements of a model variational problem for a clamped loaded beam is studied with emphasis on the effect of the beam thickness, which appears as a parameter in the problem, on the accuracy. It is shown that the approximation achieved by a standard finite element method degenerates for thin beams. In contrast a large family of mixed finite element methods are shown to yield quasioptimal approximation independent of the thickness parameter. The most useful of these methods may be realized by replacing the integrals appearing in the stiffness matrix of the standard method by Gauss quadratures.

Keywords: mixed finite element method, reduced integration, Timoshenko beam, parameter

A convergence analysis is presented for standard and mixed finite element discretizations of a model system of equations for a transversely loaded beam. The equations depend parametrically on the beam thickness and the emphasis of the analysis is on the robustness of the methods with respect to this parameter. The mixed methods are shown to be far more robust than the standard methods employing elements of the same degree. Moveover they entail no additional computational expense. Computational results are included to illustrate the main results.

Keywords: beam, mixed finite element, parameter-dependence

A standard Galerkin method for a quasilinear equation of Sobolev type using continuous, piecewise-polynomial spaces is presented and analyzed. Optimal order error estimates are established in various norms, and nodal superconvergence is demonstrated. Discretization in time by explicit single-step methods is discussed.

Keywords: superconvergence, finite element, Sobolev equation

The asympotic expansion of the Galerkin solution of a parabolic equation by means of a sequence of elliptic projections that was introduced by Douglas, Dupont, and Wheeler is carried out for a quasilinear equation. This quasi-projection can be applied to establish knot superconvergence in the case of a single space variable. In addition, an optimal order error estimate in L-infinity(L-infinity) is derived for a single space variable.

Keywords: finite element, parabolic equation, superconvergence

A nonconforming finite element procedure for the solution of second order, nonlinear parabolic boundary value problems is formulated and analyzed. The finite element space consists of discontinuous piecewise polynomial functions over quite general meshes, with inter-element continuity being enforced approximately by means of penalties. Optimal order error estimates in energy and L2 norms are stated in terms of locally expressed quantities. They are proved first for a model problem and then in general.

Keywords: finite element, interior penalty