Application of volume integrals to the solution of partial differential equations

Abstract

A group of algorithms for the numerical solution of elliptic partial differential equations is presented. The differential equation is reduced to a Fredholm equation of the second kind, which in turn is discretized by means of the Nyström algorithm, leading to a dense large-scale system of linear algebraic equations. This linear system is solved by means of the generalized conjugate residual algorithm, which requires that the matrix of the system be applied to a sequence of recursively generated vectors. Since applying a dense matrix to a vector is a process of order n 2, where n is the dimension of the system, the resulting algorithms are usually not competitive with finite differences and finite elements in terms of CPU time requirements. However, it turns out that the matrices of linear systems resulting from many partial differential equations can be applied to vectors in a “fast” manner [i.e. for the cost proportional to n in some cases and to n · log ( n) or n · (log ( n)) 2 in others], resulting in extremely efficient algorithms for the solution of certain elliptic partial differential equations. The performances of several such algorithms are illustrated by numerical examples.