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IMA Journal of Numerical Analysis Advance Access originally published online on July 25, 2007
IMA Journal of Numerical Analysis 2008 28(2):225-244; doi:10.1093/imanum/drm019
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© The author 2007. Published by Oxford University Press on behalf of the Institute of Mathematics and its Applications. All rights reserved.

On the coupling of local discontinuous Galerkin and boundary element methods for non-linear exterior transmission problems

Rommel Bustinza{dagger} and Gabriel N. Gatica{ddagger}

Departamento de Ingeniería Matemática, Universidad de Concepción, Casilla 160-C, Concepción, Chile

Francisco-Javier Sayas§

Departamento de Matemática Aplicada, Universidad de Zaragoza, Centro Politécnico Superior, Marí a de Luna 3, 50018 Zaragoza, Spain

{dagger} Email: rbustinz{at}ing-mat.udec.cl

{ddagger} Email: ggatica{at}ing-mat.udec.cl

§ Email: jsayas{at}unizar.es

Received on 17 August 2006. Revised on 10 May 2007.


  Abstract

In this paper, we apply the coupling of local discontinuous Galerkin and boundary element methods to solve a class of non-linear exterior transmission problems in the plane. As a model, we consider a non-linear elliptic equation in an annular polygonal domain coupled with the Poisson equation in the surrounding unbounded exterior region. In addition, we assume discontinuous transmission conditions on the interface boundary. Our approach constitutes an extension to non-linear problems of the a priori error analysis developed recently for linear exterior transmission problems. Most of the techniques employed here are similar to the linear case but some differences appear. In particular, because of the non-homogeneous transmission conditions, the so-called numerical fluxes need to be suitably defined. We prove stability of the resulting discrete scheme with respect to a mesh-dependent norm and derive a Strang-type estimate for the associated error. Then, we apply local and global approximation properties of the discrete spaces to obtain the a priori error estimates in the energy norm.

Key Words: boundary elements; local discontinuous Galerkin; coupling; non-linear problems


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