Multi-Grid Algorithms with Applications to Elliptic Boundary Value Problems

Abstract

This work is primarily concerned with solving the large sparse linear systems which arise in connection with finite-element or finite-difference procedures for solving self-adjoint elliptic boundary-value problems. These problems can be expressed in terms of abstract variational problems on Hilbert spaces. Our (mufti-grid) schemes involve a sequence of auxiliary finite-dimensional spaces which do not have to be nested. We approximate the solution using the largest (finite-dimensional) space. These schemes are recursive in nature: they combine smoothing iterations in a space with solving one or more correction problems using smaller spaces. Under certain circumstances, the solution to a problem can be approximated well using smaller spaces. Since the smaller spaces are required to have geometrically fewer unknowns than the largest space, the savings in computation can be substantial. In fact, we prove that these procedures are optimal order under appropriate conditions. Our general theory is discretization independent and can be applied to problems which do not arise from partial differential equations. As examples, we consider three particular discretizations of variable coefficient self-adjoint second order elliptic boundary-value problems. The first is a finite-element discretization on a convex domain in two dimensions. The second is a finite-difference discretization in one dimension. The last is a finite-difference discretization on the unit square.