Well-Posedness and Finite Element Approximation of Mixed Dimensional Partial Differential Equations

TL;DR

This paper investigates the mathematical well-posedness and regularity of mixed dimensional elliptic PDEs with embedded interfaces, proposes a finite element approximation with error bounds, and introduces an efficient iterative solver validated by numerical experiments.

Contribution

It provides the first comprehensive analysis of well-posedness, regularity, finite element approximation, and iterative solution methods for mixed dimensional PDEs with embedded interfaces.

Findings

Finite element method achieves optimal error bounds.

Proposed iterative solver converges rapidly with the new preconditioner.

Numerical experiments confirm theoretical convergence rates.

Abstract

We consider a mixed dimensional elliptic partial differential equation posed in a bulk domain with a large number of embedded interfaces. In particular, we study well-posedness of the problem and regularity of the solution. We also propose a fitted finite element approximation and prove an a priori error bound. For the solution of the arising linear system we propose and analyze an iterative method based on subspace decomposition. Finally, we present numerical experiments and achieve rapid convergence using the proposed preconditioner, confirming our theoretical findings.