A Comment on Least-Squares Finite Element Methods with Minimum Regularity Assumptions

Abstract

Least-squares(LS) finite element methods are applied successfully to a wide range of problems arising from science and engineering. However, there are reservations to use LS methods for problems with low regularity solutions. In this paper, we consider LS methods for second-order elliptic problems using the minimum regularity assumption, i.e. the solution only belongs to $H^1$ space. We provide a theoretical analysis showing that LS methods are competitive alternatives to mixed and standard Galerkin methods by establishing that LS solutions are bounded by the mixed and standard Galerkin solutions.