First order least squares method with weakly imposed boundary condition for convection dominated diffusion problems

TL;DR

This paper introduces a robust first order least squares method for convection dominated diffusion problems that weakly imposes boundary conditions, providing stable and accurate solutions even on coarse, unstructured meshes.

Contribution

It presents a novel theoretical framework linking least squares and DPG methods, with a boundary condition weak imposition technique and diffusion-independent condition number.

Findings

Provides a stable L2 a priori error estimate.

Condition number of the global matrix is independent of diffusion coefficient.

Numerical experiments confirm theoretical stability and accuracy.

Abstract

We present and analyze a first order least squares method for convection dominated diffusion problems, which provides robust L2 a priori error estimate for the scalar variable even if the given data f in L2 space. The novel theoretical approach is to rewrite the method in the framework of discontinuous Petrov - Galerkin (DPG) method, and then show numerical stability by using a key equation discovered by J. Gopalakrishnan and W. Qiu [Math. Comp. 83(2014), pp. 537-552]. This new approach gives an alternative way to do numerical analysis for least squares methods for a large class of differential equations. We also show that the condition number of the global matrix is independent of the diffusion coefficient. A key feature of the method is that there is no stabilization parameter chosen empirically. In addition, Dirichlet boundary condition is weakly imposed. Numerical experiments verify our theoretical results and, in particular, show our way of weakly imposing Dirichlet boundary condition is essential to the design of least squares methods - numerical solutions on subdomains away from interior layers or boundary layers have remarkable accuracy even on coarse meshes, which are unstructured quasi-uniform.