Coupled variational formulations of linear elasticity and the DPG methodology

TL;DR

This paper introduces a coupled variational approach for linear elasticity problems using broken formulations and the DPG method, enabling stable and convergent solutions across subdomains with potential applications to various PDEs.

Contribution

It develops a novel framework for coupling broken variational formulations in linear elasticity, ensuring well-posedness and stability via the DPG methodology.

Findings

Demonstrated convergence rates in numerical examples

Proved global well-posedness of coupled formulations

Validated approach with illustrative elasticity problems

Abstract

This article presents a general approach akin to domain-decomposition methods to solve a single linear PDE, but where each subdomain of a partitioned domain is associated to a distinct variational formulation coming from a mutually well-posed family of broken variational formulations of the original PDE. It can be exploited to solve challenging problems in a variety of physical scenarios where stability or a particular mode of convergence is desired in a part of the domain. The linear elasticity equations are solved in this work, but the approach can be applied to other equations as well. The broken variational formulations, which are essentially extensions of more standard formulations, are characterized by the presence of mesh-dependent broken test spaces and interface trial variables at the boundaries of the elements of the mesh. This allows necessary information to be naturally transmitted between adjacent subdomains, resulting in coupled variational formulations which are then proved to be globally well-posed. They are solved numerically using the DPG methodology, which is especially crafted to produce stable discretizations of broken formulations. Finally, expected convergence rates are verified in two different and illustrative examples.