Stability analysis for boundary element methods for the diffusion equation (Q1078997)

The paper studies some aspects of the stability of the integration in time of the heat equation \(u_ t=\Delta u\) by means of the boundary element method. The attention is restricted to the approximation of \(\int G(x-s,t)u(s,0)ds\), one of the integrals which contribute to u(x,\(\Delta\) t). (Here G denotes the Gaussian kernel.) It is first observed that stability cannot be obtained if the integral is naively approximated by a quadrature \(\sum w_ iG(x-s_ i,\Delta t)u(s_ i,0)\), since the values \(G(x-s_ i,\Delta t)\) become unbounded as \(\Delta\) t tends to 0. It is therefore necessary to resort to product quadrature formulae and the author analyses this alternative approach.