Consider the periodic boundary value problem - u + u = x, u(0) = u(2), u'(0) = u(2). (a) Write down the analytic solution. (b)

Consider the periodic boundary value problem - u" + u = x, u(0) = u(2π), u'(0) = u(2π).
(a) Write down the analytic solution.
(b) Write down a minimization principle.
(c) Divide the interval [0, 2tc] into n = 5 equal subintervals, and let Wn denote the subspace consisting of all piecewise affine functions that satisfy the boundary conditions. What is the dimension of W"? Write down a basis.
(d) Construct the finite element approximation to the solution to the boundary value problem by minimizing the functional in part (b) on the subspace Wn Graph the result and compare with the exact solution. What is the maximal error on the interval?
(e) Repeat part (d) for n = 10, 20, and 40 subintervals, and discuss the convergence of your solutions.