Consider the boundary value problem solved in Example 11.15. Let W" be the subspace consisting of all polynomials u(x) of degree < n satisfying the boundary conditions «(0) = n(l) = 0. In this project, we will try to approximate the exact solution to the boundary value problem by minimizing the functional (11.176) on the polynomial subspace W".
(a) For each n = 5. 10 and 20:
(b) First, determine a basis for Wn.
(c) Set up the minimization problem as a system of linear equations for the coefficients of the polynomial minimizer relative to your basis.
(d) Solve the polynomial minimization problem and graph the solution.
(e) Compare your "polynomial finite element" solution with the exact solution and the piecewise affine finite element solution graphed in Figure 11.23. Discuss what you observe.