The mortar boundary element method

Abstract

This thesis is primarily concerned with the mortar boundary element method (mortar BEM). The mortar finite element method (mortar FEM) is a well established numerical scheme for the solution of partial differential equations. In simple terms the technique involves the splitting up of the domain of definition into separate parts. The problem may now be solved independently on these separate parts, however there must be some sort of matching condition between the separate parts. Our aim is to develop and analyse this technique to the boundary element method (BEM). The first step in our journey towards the mortar BEM is to investigate the BEM with Lagrangian multipliers. When approximating the solution of Neumann problems on open surfaces by the Galerkin BEM the appropriate boundary condition (along the boundary curve of the surface) can easily be included in the definition of the spaces used. However, we introduce a boundary element Galerkin BEM where we use a Lagrangian multiplier to incorporate the appropriate boundary condition in a weak sense. This is the first step in enabling us to understand the necessary matching conditions for a mortar type decomposition. We next formulate the mortar BEM for hypersingular integral equations representing the elliptic boundary value problem of the Laplace equation in three dimensions (with Neumann boundary condition). We prove almost quasi-optimal convergence of the scheme in broken Sobolev norms of order 1/2. Sub-domain decompositions can be geometrically non-conforming and meshes must be quasi-uniform only on sub-domains. We present numerical results which confirm and underline the theory presented concerning the BEM with Lagrangian multipliers and the mortar BEM. Finally we discuss the application of the mortaring technique to the hypersingular integral equation representing the equations of linear elasticity. Based on the assumption of ellipticity of the appearing bilinear form on a constrained space we prove the almost quasi-optimal convergence of the scheme.