An Optimally Convergent Coupling Approach for Interface Problems Approximated with Higher-Order Finite Elements

TL;DR

This paper introduces a novel virtual interface coupling method for interface problems that achieves optimal convergence rates with higher-order finite elements, even on non-matching meshes, by enforcing extended interface conditions.

Contribution

The paper presents a virtual interface framework that ensures optimal convergence for interface problems with higher-order finite elements, bypassing geometric matching requirements.

Findings

Achieves optimal H^1 and L^2 convergence rates.

Works on non-matching polytopial meshes.

Ensures well-posedness and stability of the method.

Abstract

In this paper, we present a new numerical method for determining the numerical solution of interface problems to optimal accuracy with respect to the polynomial order of the Lagrangian finite element space on polytopial meshes. We introduce the notion of a virtual interface, and on this virtual interface we enforce that "extended" interface conditions are satisfied in the sense of a Dirichlet--Neumann coupling. The virtual interface framework serves to bypass geometric variational crimes incurred by the classical finite element method. Further, this approach does not require that the geometric interfaces are spatially matching. Our analysis indicates that this approach is well--posed and optimally convergent in $H^1$. Numerical experiments indicate that optimal $H^1$ and $L^2$ convergence is achieved.