DISCRETIZATION AND SOLUTION OF ELLIPTIC PDES - A DIGITAL SIGNAL-PROCESSING APPROACH

A digital signal processing (DSP) approach is used to study numerical methods for discretizing and solving linear elliptic partial differential equations (PDEs). Whereas conventional PDE analysis techniques rely on matrix analysis and on a space-domain point of view to study the performance of solution methods, the DSP approach described here relies on frequency domain analysis and on multidimensional DSP techniques. This tutorial paper discusses both discretization schemes and solution methods. In the area of discretization, mode-dependent finite-difference schemes for general second-order elliptic PDEs are examined, and are illustrated by considering the Poisson, Helmholtz, and convection-diffusion equations as examples. In the area of solution methods, we focus on methods applicable to self-adjoint positive definite elliptic PDEs. Both direct and iterative methods are discussed, which include fast Poisson solvers, elementary and accelerated relaxation methods, multigrid methods, preconditioned conjugate gradient methods and domain decomposition techniques. In addition to describing these methods in a DSP setting, an up-to-date survey of recent developments is also provided. © 1990 IEEE