Semi-discrete and fully discrete weak Galerkin finite element methods for a quasistatic Maxwell viscoelastic model

TL;DR

This paper develops semi-discrete and fully discrete weak Galerkin finite element methods for a quasistatic Maxwell viscoelastic model, providing theoretical analysis and numerical validation of their accuracy and stability.

Contribution

It introduces novel weak Galerkin finite element schemes with specific polynomial degrees for stress and velocity, and proves their well-posedness and optimal error estimates.

Findings

Proved existence and uniqueness of solutions for the schemes

Derived optimal a priori error estimates

Numerical examples confirm theoretical results

Abstract

This paper considers weak Galerkin finite element approximations for a quasistatic Maxwell viscoelastic model. The spatial discretization uses piecewise polynomials of degree $k \ (k\geq 1)$ for the stress approximation, degree $k+1$ for the velocity approximation, and degree $k$ for the numerical trace of velocity on the inter-element boundaries. The temporal discretization in the fully discrete method adopts a backward Euler difference scheme. We show the existence and uniqueness of the semi-discrete and fully discrete solutions, and derive optimal a priori error estimates. Numerical examples are provided to support the theoretical analysis.