A mixed approach to the Poisson problem with line sources. (English) Zbl 1466.35093
Summary: In this work we consider the dual-mixed variational formulation of the Poisson equation with a line source. The analysis and approximation of this problem is nonstandard, as the line source causes the solutions to be singular. We start by showing that this problem admits a solution in appropriately weighted Sobolev spaces. Next, we show that given some assumptions on the problem parameters, the solution admits a splitting into higher- and lower-regularity terms. The lower-regularity terms are here explicitly known and capture the solution singularities. The higher-regularity terms, meanwhile, are defined as the solution of an associated mixed Poisson equation. With the solution splitting in hand, we then define a singularity removal-based mixed finite element method in which only the higher-regularity terms are approximated numerically. This method yields a significant improvement in the convergence rate when compared to approximating the full solution. In particular, we show that the singularity removal-based method yields optimal convergence rates for lowest-order Raviart-Thomas and discontinuous Lagrange elements.
MSC:
| 35J05 | Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation |
| 35J75 | Singular elliptic equations |
| 65M60 | Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs |
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