Optimal adaptivity for BEM and FEM-BEM coupling

The ultimate goal of adaptive mesh-refining schemes is to compute a discrete solution with error below a prescribed tolerance at the expense of, up to a multiplicative constant, the minimal computational cost. Although adaptive strategies are successfully employed since the eighties, only comparably little of the empirical observations can be predicted and guaranteed by mathematical analysis. The proposed research aims at the further mathematical foundation of certain aspects of adaptive algorithms, with the focus on convergence and optimal convergence rates for boundary element based methods for second-order elliptic PDEs. First, we aim to consider adaptive BEM (boundary element method) for second-order elliptic PDEs. We will generalize recent own work from the lowest-order case and piecewise flat boundaries to higher polynomials on piecewise smooth boundaries and beyond the Laplace equation. Moreover, we shall include the adaptive approximation of the given data into the developed algorithm, so that a possible implementation has to deal with discrete integral operators only. We expect to prove convergence with optimal rate, where the possible rate is determined by the regularity of the given data and the (unknown) solution only. A major issue will then be to include matrix compression techniques from the field of so-called fast BEM into the convergence and quasi-optimality analysis. Second, we consider the adaptive coupling of FEM (finite element method) and BEM for different coupling formulations. Recent own work proves plain convergence by means of the estimator reduction principle, where in the presence of certain nonlinearities in the FEM domain the analysis is restricted to lowest-order FEM. We aim at the proof of optimal convergence rates also for higher- order elements. The fact that even in the linear case the operator matrices of the FEM-BEM couplings are not symmetric and/or not elliptic prevents to use techniques from the literature. Instead, new mathematical techniques have to be developed. In particular, we intend to get new insight into or even a general mathematical frame of quasi-optimal convergence of adaptive FEM in the presence of weak nonlinearities as, e.g., strongly monotone operators. Third, since an adaptive mesh-refinement allows the improved resolution of the problem geometry, we will focus on the adaptive geometry approximation which is so far essentially neglected in the literature. The development of a convergent adaptive algorithm which includes the adaptive resolution of the geometry, will have an important impact on adaptive BEM as well as FEM. The proof of optimal convergence rates in such a frame would be a fundamental milestone in the mathematical understanding of adaptive schemes.

For two reasons, fast and accurate error estimation plays a key role in reliable and efficient scientific computing: First, one may want to check whether the solution of a numerical simulation is accurate enough. Second, if this is not the case, one aims to improve the discretization, e.g., by local refinement of the underlying mesh. Both subjects are usually covered by so-called a posteriori error estimates and related adaptive mesh-refining algorithms. For error control in finite element methods, there is a broad variety of a posteriori error estimators available, and convergence as well as optimality of adaptive algorithms is well studied in the literature. For integral equations as well as the coupling of integral and differential equations, the funded research project derived a posteriori error estimates and developed and analyzed optimal adaptive mesh-refinement strategies.