On the DPG method for Signorini problems

TL;DR

This paper develops and analyzes discontinuous Petrov-Galerkin methods with optimal test functions for Signorini problems, demonstrating their effectiveness and robustness, especially in singularly perturbed cases.

Contribution

It introduces new DPG formulations for Signorini problems, including symmetric and non-symmetric variants, with error estimates and applicability to reaction-dominated diffusion.

Findings

Optimal test functions improve solution accuracy.

Method is robust in singularly perturbed scenarios.

Numerical results confirm theoretical advantages.

Abstract

We derive and analyze discontinuous Petrov-Galerkin methods with optimal test functions for Signorini-type problems as a prototype of a variational inequality of the first kind. We present different symmetric and non-symmetric formulations where optimal test functions are only used for the PDE part of the problem, not the boundary conditions. For the symmetric case and lowest order approximations, we provide a simple a posteriori error estimate. In a second part, we apply our technique to the singularly perturbed case of reaction dominated diffusion. Numerical results show the performance of our method and, in particular, its robustness in the singularly perturbed case.