A unified analysis of a weighted least squares method for first-order systems

Abstract

A unified analysis of a weighted least squares finite element method (WLSFEM) for approximating solutions of a large class of first-order differential systems is proposed. The method exhibits several advantageous features. For example, the trial and test functions are not required to satisfy the boundary conditions. Its discretization results in symmetric and positive definite algebraic systems with condition number O( h −2 + w 2). And a single piecewise polynomial finite element space may be used for all test and trial functions. Asymptotic convergence of the least squares approximations with suitable weights is established in a natural norm without requiring extra smoothness of the solutions. If, instead, the solutions are sufficiently regular, a priori error estimates can be derived under two suitable assumptions which are related respectively to the symmetric positive systems of Friedrichs and first-order Agmon-Douglis-Nirenberg (ADN) elliptic systems. Numerous model problems fit into these two important systems. Some selective examples are examined and verified in the unified framework.