Semi and fully-discrete analysis of lowest-order nonstandard finite element methods for the biharmonic wave problem

TL;DR

This paper analyzes lowest-order nonstandard finite element methods for the biharmonic wave problem, providing stability, error estimates, and numerical validation for both explicit and implicit time schemes.

Contribution

It introduces a modified Ritz projection for error analysis and establishes optimal error estimates for various discretization schemes of the biharmonic wave equation.

Findings

Optimal error estimates are proven for semidiscrete and fully discrete schemes.

Numerical experiments validate theoretical error bounds.

Stability results are established for explicit and implicit time discretizations.

Abstract

This paper discusses lowest-order nonstandard finite element methods for space discretization and explicit and implicit schemes for time discretization of the biharmonic wave equation with clamped boundary conditions. A modified Ritz projection operator defined on $H^2_0(\Omega)$ ensures error estimates under appropriate regularity assumptions on the solution. Stability results and error estimates of optimal order are established in suitable norms for the semidiscrete and explicit/implicit fully-discrete versions of the proposed schemes. Finally, we report on numerical experiments using explicit and implicit schemes for time discretization and Morley, discontinuous Galerkin, and {C$^0$ interior} penalty schemes for space discretization, that validate the theoretical error estimates.