Gradient Schemes for Linear and Non-linear Elasticity Equations

TL;DR

This paper extends the Gradient Scheme framework to elasticity equations, offering unified error analysis and convergence results for both linear and non-linear elasticity models, including methods with limited regularity.

Contribution

It adapts the Gradient Scheme framework to elasticity, providing a unified analysis and convergence results for various classical and modern numerical methods.

Findings

Error estimates for linear elasticity

Convergence results for non-linear elasticity

Embedding of classical methods in the Gradient Scheme framework

Abstract

The Gradient Scheme framework provides a unified analysis setting for many different families of numerical methods for diffusion equations. We show in this paper that the Gradient Scheme framework can be adapted to elasticity equations, and provides error estimates for linear elasticity and convergence results for non-linear elasticity. We also establish that several classical and modern numerical methods for elasticity are embedded in the Gradient Scheme framework, which allows us to obtain convergence results for these methods in cases where the solution does not satisfy the full $H^2$-regularity or for non-linear models.