Numerical Methods to Solve 2-D and 3-D Elliptic Partial Differential Equations Using Matlab on the Cluster maya

Discretizing the elliptic Poisson equation with homogeneous Dirichlet boundary conditions by the finite difference method results in a system of linear equations with a large, sparse, highly structured system matrix. It is a classical test problem for comparing the performance of direct and iterative linear solvers. We compare in this report Gaussian elimination applied to a dense system matrix, Gaussian elimination applied to a sparse system matrix, the classical iterative methods of Jacobi, Gauss-Seidel, and SOR, and finally, the conjugate gradient method without preconditioning, and the conjugate gradient method with SSOR preconditioning. The key conclusions are: (i) The comparison of dense and sparse storage shows the crucial importance of sparse storage mode to solve problems even of intermediate size. (ii) The conjugate gradient method outperforms the classical iterative methods in all cases. (iii) Preconditioning can speed up the conjugate gradient method by an order of magnitude. (iv) We find that in two dimensions Gaussian elimination of a sparse system matrix is the fastest method, but runs out of memory eventually, where iterative methods can still solve the problem, but at the price of possibly extremely long run times. (v) However, in three dimensions, the iterative methods can be significantly faster than Gaussian elimination and can solve significantly larger problems. This explains the importance of iterative methods for three-dimensional problems.