Higher Order Cut Finite Elements for the Wave Equation

TL;DR

This paper develops higher order immersed finite element methods for the wave equation, demonstrating improved convergence and stability through penalty stabilization, with analysis of condition numbers and time step restrictions.

Contribution

It introduces stabilized higher order immersed finite elements for the wave equation, ensuring stability and convergence even with small boundary cuts.

Findings

Higher order convergence achieved with stabilization.

Condition numbers are independent of boundary cuts.

Time step restrictions are comparable to standard methods.

Abstract

The scalar wave equation is solved using higher order immersed finite elements. We demonstrate that higher order convergence can be obtained. Small cuts with the background mesh are stabilized by adding penalty terms to the weak formulation. This ensures that the condition numbers of the mass and stiffness matrix are independent of how the boundary cuts the mesh. The penalties consist of jumps in higher order derivatives integrated over the interior faces of the elements cut by the boundary. The dependence on the polynomial degree of the condition number of the stabilized mass matrix is estimated. We conclude that the condition number grows extremely fast when increasing the polynomial degree of the finite element space. The time step restriction of the resulting system is investigated numerically and is concluded not to be worse than for a standard (non-immersed) finite element method.