Generalized Time Integration Schemes for Space-Time Moving Finite Elements

TL;DR

This paper introduces a moving finite element method for convection-dominated PDEs, utilizing tensor-product spaces and TR-BDF2 time integration, with numerical validation of its effectiveness.

Contribution

It presents a novel moving finite element approach with discontinuous mesh reconfigurations and error analysis for time-dependent PDEs.

Findings

Numerical illustrations validate the method's efficacy.

The approach handles mesh evolution and reconfiguration effectively.

Error estimates support the method's accuracy.

Abstract

In this paper, we analyze and provide numerical illustrations for a moving finite element method applied to convection-dominated, time-dependent partial differential equations. We follow a method of lines approach and utilize an underlying tensor-product finite element space that permits the mesh to evolve continuously in time and undergo discontinuous reconfigurations at discrete time steps. We employ the TR-BDF2 method as the time integrator for piecewise quadratic tensor-product spaces, and provide an almost symmetric error estimate for the procedure. Our numerical results validate the efficacy of these moving finite elements.