Analysis of an abstract mixed formulation for viscoelastic problems

TL;DR

This paper develops an abstract framework for analyzing mixed formulations in viscoelastic problems, establishing stability and error estimates, and demonstrating the approach on specific applications with numerical validation.

Contribution

It introduces a novel abstract analysis framework for mixed formulations in viscoelasticity, including stability and error estimates, applicable to various problems.

Findings

Stable mixed formulations for viscoelastic problems.

Error estimates for finite element discretizations.

Numerical validation on Timoshenko beam and Laplace with memory.

Abstract

This study provides an abstract framework to analyze mixed formulations in viscoelasticity, in the classic saddle point form. Standard hypothesis for mixed methods are adapted to the Volterra type equations in order to obtain stability of the proposed problem. Error estimates are derived for suitable finite element spaces. We apply the developed theory to a bending moment formulation for a linear viscoelastic Timoshenko beam and for the Laplace operator with memory terms. For both problems we report numerical results to asses the performance of the methods.