Trefftz Approximation Space for Poisson Equation in Perforated Domains

Abstract

For the Poisson equation posed in a planar domain containing a large number of polygonal perforations, we propose a low-dimensional approximation space based on a coarse polygonal partitioning of the domain. Similar to other multi-scale numerical methods, this coarse space is spanned by basis functions that are locally discrete harmonic. We provide an error estimate in the energy norm that only depends on the regularity of the solution over the edges of the coarse skeleton. For a specific edge refinement procedure, this estimate allows us to establish superconvergence of the method, even if the true solution has low general regularity. Combined with the Restricted Additive Schwarz method, the proposed coarse space leads to an efficient two-level iterative linear solver which achieves the fine-scale finite element error in few iterations. The numerical experiment showcases the use of this coarse space over test cases involving singular solutions and realistic urban geometries.