Homework 10

  1. Let u(x) be the function that minimizes the integral
    ∫ 2[ ( )2 ] - x du- I[u] = 0 e dx - u dx
    among all C2 functions that satisfy the boundary conditions u(0) = 0,u(2) = 0.
    1. Write out the boundary value problem satisfied by u(x).
    2. Use the finite element method to approximate the solution. Use an equally spaced mesh with
      0 = x0 < x1 < ⋅⋅⋅ < xn+1 = 2
      with n = 4, 8, 16.
    3. By using the poor man’s method to approximate the rms error of your approximations, predict how many mesh points would be required for 6 digit accuracy (assuming no round off error)?
  2. Find the three finite element functions ωkν(x,y) associated with the triangle with vertices (0, 1), (1,-1) and (-1, 0).
  3. Write down the elemental stiffnesses for the following triangles: (a) the triangle with vertices (0, 1), (-1, 2), (0,-1); (b) a 30 - 60 - 90 degree right triangle.
  4. A metal plate is in the shape of an equilateral triangle T with vertices (-1, 0), (1, 0), and (0,√ -- 3). The edges of the triangle are fixed at temperature 0 while the center is heated with source f(x,y) = ∥(x,y) - c∥, i.e., the heat source is equal to the distance from center c = (0, 1∕√ -- 3). The steady state heat distribution u(x,y) of the triangle is given by the boundary value problem
    - Δu = f(x,y)       for (x,y) ∈ T
    u(x,y) = 0       for (x,y) ∈ ∂T
    1. Write down the minimization principle satisfied by u(x,y).
    2. In order to approximate u(x,y), the plate is divided into smaller equilateral triangles, with n triangles on each side, and the resulting finite element approximation is computed. How many triangles are in the triangulation? How many interior nodes? How many boundary nodes?
    3. For n = 6, set up and solve the finite element linear system to find an approximation to the temperature of the center of the triangle.