Numerical Methods for Elliptic PDEs

Laplace’s and Poisson’s equations are presented as our primary examples of elliptic PDEs. Numerical methods are developed first for the 2D Laplace equation, and later methods for solving the 3D Laplace equation and the 2D and 3D Poisson equation are discussed. In 1D, the banded matrix direct method is faster, but in 2D and 3D, modern iterative methods are faster. In 3D, not only are modern iterative methods much faster than banded/sparse matrix direct solvers, but iterative methods are required due to memory usage. The classical Jacobi, Gauss-Seidel, and SOR iterative methods are developed for Laplace’s equation, and then the general theory of classical iterative methods is developed. The modern approach to solving \(A x = b\) is to iteratively seek the minimum of a quadratic form. Minimization methods like PCG and GMRES are even faster and more robust than SOR. As an example of a nonlinear elliptic PDE, the 2D nonlinear Laplace equation is solved using Newton’s method and GMRES.