Finite Element Approximation of Elliptic Problems with Dirac Measure Terms in Weighted Spaces: Applications to One- and Three-dimensional Coupled Problems

Abstract

In this work we study the stability and the convergence rates of the finite element approximation of elliptic problems involving Dirac measures, using weighted Sobolev spaces and weighted discrete norms. Our approach handles both the cases where the measure is simply a right-hand side and where it represents an additional term, i.e., solution-dependent, in the formulation of the problem. The main motivation of this study is to provide a methodological tool to treat elliptic problems in fractured domains, where the coupling terms are seen as Dirac measures concentrated on the fractures. We first establish a decomposition lemma, which is our fundamental tool for the analysis of the considered problems in the nonstandard setting of weighted spaces. Then, we consider the stability of the Galerkin approximation with finite elements in weighted norms with uniform and graded meshes. We introduce a discrete decomposition lemma that extends the continuous one and allows us to derive discrete inf-sup conditions in weighted norms. Then, we focus on the convergence of the finite element method. Due to the lack of regularity, the convergence rates are suboptimal for uniform meshes; we show that using graded meshes, optimal rates are recovered. Our theoretical results are supported by several numerical experiments. Finally, we show how our theoretical results apply to certain coupled problems involving fluid flow in porous three-dimensional media with one-dimensional fractures that are found in the analysis of microvascular flows.