A Priori Error Estimates for Mixed Finite Element $\theta$-Schemes for the Wave Equation

TL;DR

This paper introduces a family of implicit mixed finite element schemes with a three-level time-stepping method for the wave equation, providing stability and optimal error estimates in the displacement and pressure variables.

Contribution

It develops a new implicit mixed finite element approach with a three-level scheme and provides rigorous stability and convergence analysis with sharp error estimates.

Findings

Stable energy estimates established for the scheme

Optimal a priori error bounds derived in L-infinity(L2) norm

Applicable to acoustic wave equation discretization

Abstract

A family of implicit-in-time mixed finite element schemes is presented for the numerical approximation of the acoustic wave equation. The mixed space discretization is based on the displacement form of the wave equation and the time-stepping method employs a three-level one-parameter scheme. A rigorous stability analysis is presented based on energy estimation and sharp stability results are obtained. A convergence analysis is carried out and optimal a priori $L^\infty(L^2)$ error estimates for both displacement and pressure are derived.