Mixed virtual element methods for a stress-velocity-rotation formulation in viscoelasticity

TL;DR

This paper introduces a novel mixed virtual element method for simulating viscoelastic materials, effectively handling stress decomposition and rotation, with proven stability and optimal error estimates validated through numerical experiments.

Contribution

A new mixed virtual element formulation for viscoelasticity that weakly imposes stress symmetry and includes a comprehensive stability and error analysis.

Findings

Proven unique solvability of semi-discrete and fully-discrete problems.

Established optimal a priori error estimates for all variables.

Numerical examples confirm theoretical accuracy and robustness.

Abstract

In this paper we propose a new mixed virtual element formulation for the numerical approximation of viscoelasticity equations with weakly imposed stress symmetry. The governing equations use the Zener model and are expressed in terms of the principal unknowns of additively decomposed stress into elastic and internal viscoelastic contributions, while the rotation tensor and velocity act as Lagrange multipliers. The time discretisation uses Crank--Nicolson's scheme. We demonstrate the unique solvability of both semi-discrete and fully-discrete problems by leveraging the properties of suitable local projectors. Moreover, we establish optimal a priori error estimates for all variables that appear in the mixed formulation. To validate our theoretical findings, we present several representative numerical examples that also highlight the features of the proposed formulation.