C1-conforming variational discretization of the biharmonic wave equation

Abstract

Biharmonic wave equations are of importance to various applications including thin plate analyses. The innovation of this work comes through the numerical approximation of their solutions by a C1-conforming in space and time finite element approach. Therein, the smoothness properties of solutions to the continuous evolution problem are embodied. 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 by numerical experiments with complex wave profiles in homogeneous and heterogeneous media, illustrating that the approach offers high potential also for sophisticated multi-physics and/or multi-scale systems.