Numerics of boundary-domain integral and integro-differential equations for BVP with variable coefficient in 3D

Abstract

This is the post-print version of the article. The official published version can be accessed from the links below - Copyright @ 2013 Springer-VerlagA numerical implementation of the direct boundary-domain integral and integro-differential equations, BDIDEs, for treatment of the Dirichlet problem for a scalar elliptic PDE with variable coefficient in a three-dimensional domain is discussed. The mesh-based discretisation of the BDIEs with tetrahedron domain elements in conjunction with collocation method leads to a system of linear algebraic equations (discretised BDIE). The involved fully populated matrices are approximated by means of the H-Matrix/adaptive cross approximation technique. Convergence of the method is investigated.This study is partially supported by the EPSRC grant EP/H020497/1:"Mathematical Analysis of Localised-Boundary-Domain Integral Equations for Variable-Coefficients\ud Boundary Value Problems"

A new family of boundary-domain integral equations for the diffusion equation with variable coefficient in unbounded domains

“…which has been used to derive analogous results to those in this paper, in [9]. The parametrix P y has been widely analysed in the literature, see [21,20,14,7,8]. The difference between both parametrices relies on the dependence from the variable of the coefficient a(x) or a(y).…”

“…In [3], the authors show that it is possible to obtain linear convergence with respect to the number of quadrature curves, and in some cases, exponential convergence. Analogous research in 3D shows the successful implementation of fast algorithms to obtain the solution of boundary domain integral equations, see [27,14,28]. Furthermore, the authors [4] show the application of the Boundary Domain Integral Equation Method to the study of inverse problems with variable coefficients.…”

Boundary-domain integral method for vorticity transport equation with variable viscosity

“…Variable diffusivity was considered by Grzhibovskis et al [8], Chkadua et al [5], Al Jawary [1][2][3] and Ang et al [4]. Ravnik and Skerget [12] proposed a boundary-domain integral formulation for diffusion-convection equations with variable coefficient and velocity and considered the energy equation with variable material properties, [13].…”

Localized boundary‐domain singular integral equations of Dirichlet problem for self‐adjoint second‐order strongly elliptic PDE systems

“…We show that a solution of the problem can be represented by explicit localized parametrix-based potentials and that the corresponding localized boundary-domain integral operator (LBDIO) is invertible, which is important for analysis of convergence and stability of localized boundary-domain integral equation (LBDIE)-based numerical methods for PDEs (e.g. [6][7][8][9][10][11][12][13]).…”