A Petrov-Galerkin method for nonlocal convection-dominated diffusion problems

TL;DR

This paper introduces a Petrov-Galerkin method with an optimal test norm for nonlocal convection-dominated diffusion problems, ensuring stability and convergence even with sharp gradients, and confirms its asymptotic compatibility.

Contribution

The paper develops a Petrov-Galerkin framework with an optimal test norm for nonlocal problems, providing stability, convergence, and asymptotic compatibility.

Findings

The method achieves uniform inf-sup stability independent of the problem.

Numerical experiments demonstrate optimal convergence with $h$- and $p$-refinements.

The approach is asymptotically compatible for nonlocal to local limits.

Abstract

We present a Petrov-Gelerkin (PG) method for a class of nonlocal convection-dominated diffusion problems. There are two main ingredients in our approach. First, we define the norm on the test space as induced by the trial space norm, i.e., the optimal test norm, so that the inf-sup condition can be satisfied uniformly independent of the problem. We show the well-posedness of a class of nonlocal convection-dominated diffusion problems under the optimal test norm with general assumptions on the nonlocal diffusion and convection kernels. Second, following the framework of Cohen et al.~(2012), we embed the original nonlocal convection-dominated diffusion problem into a larger mixed problem so as to choose an enriched test space as a stabilization of the numerical algorithm. In the numerical experiments, we use an approximate optimal test norm which can be efficiently implemented in 1d, and study its performance against the energy norm on the test space. We conduct convergence studies for the nonlocal problem using uniform $h$- and $p$-refinements, and adaptive $h$-refinements on both smooth manufactured solutions and solutions with sharp gradient in a transition layer. In addition, we confirm that the PG method is asymptotically compatible.