|Subject:||A Non-standard Finite Element Method using Boundary Integral Operators|
|Author:||Dipl.-Ing. Clemens Hofreither|
|Supervision:||O.Univ.-Prof. Dr. Ulrich Langer|
|Co-supervision:||Dr. Dipl.-Ing. Clemens Pechstein|
|External referee:||Prof. Dr. Sergej Rjasanow (Universität des Saarlandes)|
This thesis is concerned with the analysis of a relatively novel discretization scheme for boundary value problems of second-order elliptic partial differential equations. The method decomposes the computational domain into elements and uses trial functions with local support. In contrast to standard finite element method (FEM) discretizations, the mesh may consist of arbitrary polygonal or polyhedral elements, and the trial functions are not locally polynomial, but locally PDE-harmonic, by which we mean that they satisfy the partial differential equation on every element. The method may thus be considered to be in the tradition of Trefftz methods. The PDE-harmonic trial functions are constructed via the solution of element-local boundary value problems. A characteristic feature of the scheme discussed in this thesis is that these local problems are tackled using boundary integral operators, and in particular a boundary element method (BEM) discretization. For this reason, we refer to the method as a BEM-based FEM. Indeed, it can be regarded as a finite element method where the element stiffness matrices are computed using boundary element techniques.
For the construction of the involved boundary integral operators, explicit knowledge of a fundamental solution of the partial differential operator is required. However, due to the local construction, we only need local fundamental solutions for the element problems, in contrast to standard BEM approaches. Therefore, we study the setting of elementwise constant coefficients of the partial differential operator, since a fundamental solution is readily available in the literature for operators with constant coefficients.
Alternatively to the interpretation as a Trefftz method, the method may also be viewed as a variant of a domain decomposition technique using boundary integral operators. The main difference to this approach lies in the discretization strategy: in contrast to domain decomposition methods, where the subdomains are typically of moderate to large size in order to enable efficient parallel processing, we consider the substructures in the BEM-based FEM as elements with only a small number of degrees of freedom. This has ramifications for the analysis as well since we need to develop analytical tools for arbitrary polygonal or polyhedral meshes with mesh sizes which uniformly tend to zero. These estimates are typically proven using the mapping principle in the FEM literature, an approach which fails for heterogeneous polytopal meshes. Thus, new techniques have to be developed for deriving these results.
With these analytical tools at hand, the first major result of the thesis is the derivation of rigorous error estimates for the BEM-based FEM on heterogeneous polyhedral meshes for a model problem. In particular, we prove both H1- and, by passing to an equivalent mixed formulation, L2-error estimates which are quasi-optimal with respect to the approximation properties of the underlying skeletal space. Further results include the derivation and convergence analysis of an efficient parallel solver for the resulting system of linear equations which is based on the ideas of the one-level finite element tearing/interconnecting (FETI) substructuring technique. Furthermore, we consider the application of the method to convection-diffusion problems, where the use of PDE-harmonic trial functions confers a stability advantage over a standard FEM discretization. In fact, we show that the BEM-based FEM is closely related to the method of residual-free bubbles and thus also to the well-established SUPG scheme. In a final chapter, we present numerical examples in order to confirm some of the theoretical results of the thesis.
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