Finite differences and finite elements: getting to know you

T o debug your programs, it’s helpful to experiment with the simplest test problem and a small number of mesh points. Look ahead to Problem 6 for sample problems. Problem 2 uses the Matlab function spdiags to construct a sparse matrix. If you have never used sparse matrices in Matlab, print the matrix A to see that its data structure contains the row index, column index, and value for each nonzero element. If you have never used spdiags, type help spdiags to see the documentation, and then try it on your own data to see exactly how the matrix elements are defined. Use Matlab’s quad to compute the integrals for the entries in the matrix and right-hand side for the finite element formulations. Before tackling the programming for Problems 5 and 6, take some time to understand exactly where the nonzeros are in the matrix, and exactly what intervals of integration should be used. The programs are short, but it’s easy to make mistakes if you don’t understand what they compute. In Problem 7, we measure work by counting the number of multiplications. One alternative is to count the number of floating-point computations, but this usually gives a count of about twice the number of multiplications, because multiplications and additions are typically paired in computations. Computing time is another very useful measure of work, but it can be contaminated by the effects of other users or computer processes. In determining and understanding the convergence rate in Problem 7, plotting the solutions or the error norms might be helpful. Mark Gockenbach gives a good introduction to the theory of finite difference and finite element methods;1 for a more advanced treatment, see, for example, Stig Larsson and Vidar Thomée’s book.2