Numerical Methods Qualification Exam Problems and Solutions (University of Maryland)/Aug09 667

Problem 4 Given the two-point boundary value problem Problem 4a Set up the finite element approximation for this problem, based on piecewise linear elements in equidistant points. Determine the convergence rate in an appropriate norm Solution 4a Let Find such that for all or after integrating by parts and including initial conditions Discrete Variational Form piecewise linear is basis for ; For Find such that for all Since forms a basis Therefore we have system of equations For Convergence Rate In general terms, we can use Cea's Lemma to obtain In particular, we can consider as the Lagrange interpolant, which we denote by . Then, . It's easy to prove that the finite element solution is nodally exact. Then it coincides with the Lagrange interpolant, and we have the following punctual estimation: Problem 4b Explain whether is necessary for the convergence in part (a). Solution 4b If , then the stiffness matrix is diagonally dominant and hence solvable. Solution 4 Problem 5 Solution 5 Problem 6 Solution 6