The exact solution to the boundary value problem
- u" = 3x, u(0) = u(1) = 0, is u(x) = u(x)- 1/2x-1/2x3.
(a) Use the finite element method based on 5 equally spaced nodes to approximate the solution.
(b) Compare the graphs of the exact solution and its piecewise affine finite element approximation. What is the maximal error
(i) at the mesh points?
(ii) on the entire interval?
(c) Suppose you employ a natural cubic spline to interpolate the sample values of the finite element solution at the nodes. How does that affect the errors in part (c)?