A boundary integral equation method for the numerical solution of a second order elliptic equation with variable coefficients

Abstract

A method is derived for the solution of boundary value problems governed by a second-order elliptic partial differential equation with variable coefficients. The method is obtained by expressing the solution to a particular problem in terms of an integral taken round the boundary of the region under consideration.

Fast multipole preconditioners for sparse matrices arising from elliptic equations

“…Although analytic fundamental solutions can sometimes be found for problems with variable diffusion (see, e.g., Cheng [17] and Clements [18]), in most cases numerical techniques are employed. One popular method is to introduce a number of subdomains, on each of which the diffusion coefficient is approximated by a constant function [49,67].…”

Numerical solution of a linear elliptic partial differential equation with variable coefficients: A complex variable boundary element approach

“…To deal with the domain integral in an effective manner or to obtain alternative formulations that do not require the solution domain to be discretized, various approaches have been proposed in the literature. For example, Rangogni [2] considered the case in which γ ij = δ ij (Kronecker-delta) and g(x 1 , x 2 ) = 1 + g 0 (x 1 , x 2 ) and employed the boundary element method together with the perturbation technique to solve the boundary value problem for small parameter ; Clements [3] and Ang et al [4] derived special fundamental solutions for the case in which γ ij = δ ij and g(x 1 , x 2 ) = X(x 1 )Y (x 2 ); Kassab and Divo [5] introduced the idea of a generalized fundamental solution; Tanaka et al [6] and Ang et al [7] applied the dual-reciprocity method proposed by Brebbia and Nardini [8] to approximate the domain integral as a boundary integral.…”

“…For the special case in which γ ij = δ ij and g(x 1 , x 2 ) = X(x 1 )Y (x 2 ), solutions of (1) can be expressed as a series containing an arbitrary complex function which is holomorphic in R (Clements [3] and Ang et al [4]). In Park and Ang [12] and Ang et al [13], the Cauchy integral formulae are employed to obtain a complex variable boundary element method for constructing the complex function that satisfies the boundary condition in (2).…”

A boundary‐element method for the solution of a class of time‐dependent problems for inhomogeneous media

“…where Fo(x, y) is a solution to (1 1 ) with v = kw2. Thus if p(x) is given by p ( x ) = ( a x + P)p and p ( x ) and v are given, respectively, by (19) and (20), then (18) is certainly a solution to (6).…”