A fitted space-time finite element method for an advection-diffusion problem with moving interfaces

TL;DR

This paper develops a space-time finite element method tailored for advection-diffusion problems with moving interfaces, effectively handling discontinuities and providing optimal error estimates validated by numerical experiments.

Contribution

It introduces a novel interface-fitted finite element scheme for nonstationary interfaces with discontinuous coefficients, including rigorous analysis and error estimates.

Findings

Proves well-posedness of the variational problem.

Establishes optimal error estimates in a discrete energy norm.

Numerical results confirm theoretical accuracy and stability.

Abstract

This paper presents a space-time interface-fitted finite element method for solving a parabolic advection-diffusion problem with a nonstationary interface. The jumping diffusion coefficient gives rise to the discontinuity of the solution gradient across the interface. We use the Banach-Necas-Babuska theorem to show the well-posedness of the continuous variational problem. A fully discrete finite-element based scheme is analyzed using the Galerkin method and unstructured interface-fitted meshes. An optimal error estimate is established in a discrete energy norm under a globally low but locally high regularity condition. Some numerical results corroborate our theoretical results.