$C^1$-conforming variational discretization of the biharmonic wave equation

TL;DR

This paper introduces a $C^1$-conforming finite element method for solving biharmonic wave equations, providing optimal error estimates and demonstrating its effectiveness through numerical experiments.

Contribution

It presents a novel $C^1$-conforming discretization approach for biharmonic wave equations with proven optimal error estimates.

Findings

Optimal order error estimates established

Numerical experiments confirm convergence and performance

Potential for complex multi-physics applications

Abstract

Biharmonic wave equations are of importance to various applications including thin plate analyses. In this work, the numerical approximation of their solutions by a $C^1$-conforming in space and time finite element approach is proposed and analyzed. Therein, the smoothness properties of solutions to the continuous evolution problem is embodied. High potential of the presented approach for more sophisticated multi-physics and multi-scale systems is expected. Time discretization is based on a combined Galerkin and collocation technique. For space discretization the Bogner--Fox--Schmit element is applied. Optimal order error estimates are proven. The convergence and performance properties are illustrated with numerical experiments.