A Velocity-based Moving Mesh Virtual Element Method

TL;DR

This paper introduces a velocity-based virtual element method for moving mesh PDEs, enabling accurate and flexible mesh adaptation on polygonal grids with easy node insertion for complex boundary movements.

Contribution

It extends the virtual element method to moving meshes, allowing accurate solution transfer and node insertion on polygonal meshes in PDE problems with moving boundaries.

Findings

Achieves same accuracy as linear finite element methods on polygonal meshes.

Enables easy node insertion on mesh edges for boundary adaptation.

Demonstrates effective handling of boundary interactions with solid objects.

Abstract

We present a velocity-based moving mesh virtual element method for the numerical solution of PDEs involving moving boundaries. The virtual element method is used for computing both the mesh velocity and a conservative Arbitrary Lagrangian-Eulerian solution transfer on general polygonal meshes. The approach extends the linear finite element method to polygonal mesh structures, achieving the same degree of accuracy. In the context of moving meshes, a major advantage of the virtual element approach is the ease with which nodes can be inserted on mesh edges. Demonstrations of node insertion techniques are presented to show that moving polygonal meshes can be simply adapted for situations where a boundary encounters a solid object or another moving boundary, without reduction in degree of accuracy.