Efficient Quadrature Settings for Elliptic PDE’s using a Coupled FEM and BEM Solver in COMSOL Multiphysics

Sammanfattning: By using singular integral kernels based on the fundamental solution, a partial differential equation (PDE) can be rewritten as a boundary integral defined at the boundary of a domain. This requires a linear differential operator with coefficients that are isotropic and homogeneous in space. In this report, emphasis is put on PDE’s related to electromagnetics i.e., Laplace’s and Helmholtz equation. Both three- and two-dimensional model problems will be investigated. Galerkin’s method is implemented in order to discretize the domain, now with one less spatial dimension. Hence, the solution is expanded in a series of shape functions 'I whereafter the equation is multiplied by a test function and integrated over the boundary. The resulting matrix elements are double integrals between one shape function 'I and one test function vj integrating vertex i and j in the generated mesh. Unlike a strict BEM implementation, this report will cover a coupled BEM and FEM solver using Costabel’s Symmetric Coupling. Hence, the resulting system of equations, represented by the stiffness matrix K, consists of both sparse and dense parts originating from the different methods. FEM is usually defined in a domain where there exist non-linearities and BEM is implemented at its boundary in order to simulate an infinite domain as efficiently as possible. Furthermore, the integrals in K are transformed using two coordinate transforms: one to the reference element and another to avoid the singularity due to the integral kernel. The latter is modified for each case of integration, namely same elements, same edge, same vertex, close elements and distant elements. The objective of this report is to investigate how the settings for the numerical integration i.e., the quadrature corresponding to the different cases, affect the accuracy of the final solutions to the given PDE’s. However, an element in K is an integral of a function S which characteristics depend on several things, namely the order of the shape functions, the integral kernel and the element order of the mesh. In order to facilitate the error estimation, the numerical results will be generated from the model problems where the analytical solution is known. An efficient quadrature is achieved when the error originating from the numerical integration of S is small or neglected in comparison to the truncation errors i.e., errors originating from meshing and discretization. The thesis is written in close collaboration with the Swedish software company COMSOL Multiphysics®, thus all numerical results will be generated from this software using version 4.4.