A Novel Least-Squares Finite Element Method Enriched with Residual-Free Bubbles for Solving Convection-Dominated Problems

Abstract

In this paper we devise a novel least-squares finite element method (LSFEM) for solving scalar convection-dominated convection-diffusion problems. First, we convert a second-order convection-diffusion problem into a first-order system formulation by introducing the gradient $\mathbf{p}:=-\kappa\nabla u$ as an additional variable. The LSFEM using continuous piecewise linear elements enriched with residual-free bubbles for both variables $u$ and $\mathbf{p}$ is applied to solve the first-order mixed problem. The residual-free bubble functions are assumed to strongly satisfy the associated homogeneous second-order convection-diffusion equations in the interior of each element, up to the contribution of the linear part, and vanish on the element boundary. To implement this two-level least-squares approach, a stabilized method of Galerkin/least-squares type is used to approximate the residual-free bubble functions. This enriched LSFEM not only inherits the advantages of the primitive LSFEM, such as the resulting linear system being symmetric and positive definite, but also exhibits the characteristics of the residual-free bubble method without involving stability parameters. Several numerical experiments are given to demonstrate the effectiveness of the proposed enriched LSFEM. The accuracy and computational cost of this enriched LSFEM are also compared with those of the primitive LSFEM. We find that for a small diffusivity $\kappa$, the enriched LSFEM is much better able to capture the nature of layer structure in the solution than the primitive LSFEM, even if the primitive LSFEM uses a very fine mesh or higher-order elements. In other words, the enriched LSFEM provides a significant improvement, with a lower computational cost, over the primitive LSFEM for solving convection-dominated problems.