Numerical solution of the generalized Laplace equation by coupling the boundary element method and the perturbation method
The generalized Laplace equation of the title is \(\nabla.(a\nabla v)=0\) which is to be solved in a bounded region D of \({\mathbb{R}}^ 2\) subject to mixed boundary conditions. It is assumed that \(a(x,y)>0\) in D. Set \(v=u.a^{-}\), then u satisfies an equation in D of the form \(\nabla^ 2u+fu=0.\) The proposal is to solve the equation \(\nabla^ 2\omega +\epsilon f\omega =0\) by expanding its solution as \(\sum^{\infty}_{j=0}\epsilon^ ju_ j.\) This leads to an infinite sequence of coupled boundary value problems for \(u_ 0,u_ 1,...\), each of which will satisfy a Poisson equation. The required solution should be \(\sum^{\infty}_{j=0}u_ j.\) There is a brief discussion of the numerical methods which were used to solve the Poisson equations. The vital problem of the convergence of the solution series is not considered. The author comments however that for the two worked examples for which the solution is known only two terms of the series sufficed. The results are not wholly convincing.
scientific article; zbMATH DE number 4013951