Approximate the solution to the partial differential equation


subject to the Dirichlet boundary condition
u(x, y) = 0,
using the Finite-Element Algorithm 12.5 with the elements given in the accompanying figure. Compare the approximate solution to the actual solution,
u(x, y) = sin 5π / 2 x sin 5π / 2 y,
at the interior vertices and at the points (0.125, 0.125), (0.125, 0.25), (0.25, 0.125), and (0.25, 0.25).