Semi-discrete and fully discrete mixed finite element methods for Maxwell viscoelastic model of wave propagation

TL;DR

This paper develops and analyzes semi-discrete and fully discrete mixed finite element methods for wave propagation in Maxwell-viscoelastic models, providing error estimates, stability results, and numerical verification.

Contribution

It introduces a mixed finite element framework for Maxwell-viscoelastic wave problems, including a Crank-Nicolson scheme and comprehensive error and stability analysis.

Findings

Error estimates for semi-discrete and fully discrete schemes

Unconditional stability of the fully discrete scheme

Numerical experiments confirming theoretical results

Abstract

Semi-discrete and fully discrete mixed finite element methods are considered for Maxwell-model-based problems of wave propagation in linear viscoelastic solid. This mixed finite element framework allows the use of a large class of existing mixed conforming finite elements for elasticity in the spatial discretization. In the fully discrete scheme, a Crank-Nicolson scheme is adopted for the approximation of the temporal derivatives of stress and velocity variables. Error estimates of the semi-discrete and fully discrete schemes, as well as an unconditional stability result for the fully discrete scheme, are derived. Numerical experiments are provided to verify the theoretical results.