Functional Error Estimates for PDEs on Unbounded Domains

• Applicant: Professor Dr. Dirk Pauly
• Subject Area: Mathematics
• Term: since 2023
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 530764468

o Wider research context. In computational PDEs, the convergence of numerical schemes is spoiled by singularities of the given data and/or the unknown solution. Moreover, PDEs on unbounded domains often require boundary elements methods (BEM) involving dense matrices. With the ultimate goal to compute a discrete solu- tion with error below a prescribed tolerance at quasi-minimal computational cost, the numerical scheme has to balance the discretization error, the consistency error (stemming from matrix approximation), and the solver error of an iterative solver. o Methods and objectives. A recent work of the applicants proposes functional a-posteriori error estimates for the Laplace problem, which control the potential error of the PDE solution instead of the error of the computed integral density. The advantage of this approach for engineering is that it focuses on physical quantities and also covers collocation BEM. The project aims can roughly be summarized as follows: (1) While the mentioned own paper covers BEM for the Laplace problem on bounded domains, numerical experiments (and algorithms) have only been developed for 2D. We aim to develop, test, and validate a 3D implementation. (2) We extend the analysis of functional error estimates to Laplace problems on unbounded exterior domains and to Laplace-type transmission problems with (possibly) strongly monotone nonlinearity that are discretized by FEM-BEM couplings.(3) We address stationary 3D Maxwell equations. The latter are mathematically challenging, since infinite- dimensional kernels have to be handled by sophisticated operator theoretical methods and numerically lead to multiple saddle point formulations. (4) We develop adaptive strategies, which include the compression of the involved BEM matrices and the adaptive termination of the iterative solver. Moreover, we aim for rigorous mathematical convergence results – at least for Galerkin BEM.o Innovation. We will provide mathematical understanding of the optimal interplay of adaptive mesh-refinement, iterative solvers, and BEM matrix compression. While of strong need in practice, the analysis on this is scarce (beyond asymptotic results on uniform meshes). For the Maxwell equations, the project will make important contributions with respect to a-posteriori error estimation for quantities of physical interest on unbounded domains. All theoretical findings will be implemented in MATLAB (in 2D) and NGSolve/BEM++ (in 3D). The codes will be provided to the academic public.

• DFG Programme: Research Grants
• International Connection: Austria
• Cooperation Partner: Professor Dr. Dirk Praetorius