A coarse preconditioner for multi-domain boundary element equations

Abstract

This paper is concerned with the application of a coarse preconditioner, the generalised minimal residual (GMRES) method and a generalised successive over-relaxation (GSOR) method to linear systems of equations that are derived from boundary integral equations. Attention is restricted to systems of the form ∑ N j=1 H ij x j = c i , i=1,2,…, N, where H ij are matrices, x j and c i are column vectors. The integer N denotes the number of domains and these systems are solved by adapting techniques initially devised for solving single-domain problems. These techniques include parameter matrix accelerated GMRES and GSOR in combination with a multiplicative Schwarz method for non-overlapping domains. The multiplicative Schwarz method is a generalised form of the block Gauss–Seidel method and is called the generalised multi-domain iterative procedure. A new form of coarse grid preconditioning is applied to limit the convergence dependence on block numbers. The coarse preconditioner is obtained from a crude representation of the global system of equations. Attention is restricted to thermal problems with domains connected through resistive thermal barriers. The effect of lowering and increasing the thermal resistance between domains is investigated. The coarse preconditioner requires a more accurate representation on interfaces with lower thermal resistance. Computation times are determined for the iterative procedures and for elimination techniques indicating the relative benefits for problems of this nature.