A multi-mesh finite element method for Lagrange elements of arbitrary degree

TL;DR

This paper introduces a multi-mesh finite element method allowing independent mesh adaptation for different variables in nonlinear, time-dependent PDEs, reducing computational costs and easily integrating into existing codes.

Contribution

It presents a novel multi-mesh approach for Lagrange finite elements of arbitrary degree, applicable in multiple dimensions, with demonstrated efficiency in various complex simulations.

Findings

Reduces computational runtime by over 50% in examples

Applicable to 2D and 3D problems including fluid dynamics

Easily integrated into existing finite element software

Abstract

We consider within a finite element approach the usage of different adaptively refined meshes for different variables in systems of nonlinear, time-depended PDEs. To resolve different solution behaviours of these variables, the meshes can be independently adapted. The resulting linear systems are usually much smaller, when compared to the usage of a single mesh, and the overall computational runtime can be more than halved in such cases. Our multi-mesh method works for Lagrange finite elements of arbitrary degree and is independent of the spatial dimension. The approach is well defined, and can be implemented in existing adaptive finite element codes with minimal effort. We show computational examples in 2D and 3D ranging from dendritic growth to solid-solid phase-transitions. A further application comes from fluid dynamics where we demonstrate the applicability of the approach for solving the incompressible Navier-Stokes equations with Lagrange finite elements of the same order for velocity and pressure. The approach thus provides an easy to implement alternative to stabilized finite element schemes, if Lagrange finite elements of the same order are required.