Adaptive Least-Squares Methods for Convection-Dominated Diffusion-Reaction Problems

TL;DR

This paper develops adaptive least-squares finite element methods for convection-dominated diffusion-reaction problems, providing theoretical error estimates and demonstrating improved accuracy and efficiency through adaptive mesh refinement.

Contribution

It introduces adaptive least-squares methods based on the first-order system, with proven coercivity and error estimates, enhancing computational accuracy for convection-dominated problems.

Findings

All methods achieve the same convergence rate with sufficiently fine meshes.

Adaptive methods improve accuracy and reduce computational cost.

Numerical results validate theoretical error estimates.

Abstract

This paper studies adaptive least-squares finite element methods for convection-dominated diffusion-reaction problems. The least-squares methods are based on the first-order system of the primal and dual variables with various ways of imposing outflow boundary conditions. The coercivity of the homogeneous least-squares functionals are established, and the a priori error estimates of the least-squares methods are obtained in a norm that incorporates the streamline derivative. All methods have the same convergence rate provided that meshes in the layer regions are fine enough. To increase computational accuracy and reduce computational cost, adaptive least-squares methods are implemented and numerical results are presented for some test problems.