Adaptive First-Order System Least-Squares Finite Element Methods for Second-Order Elliptic Equations in Nondivergence Form

Abstract

This paper studies adaptive first-order system least-squares finite element methods (LSFEMs) for second-order elliptic partial differential equations in nondivergence form. Unlike the classical finite element methods which use weak formulations of PDEs that are not applicable for the nondivergence equation, the first-order least-squares formulations naturally have stable weak forms without using integration by parts, allow simple finite element approximation spaces, and have built-in a posteriori error estimators for adaptive mesh refinements. The nondivergence equation is first written as a system of first-order equations by introducing the gradient as a new variable. Then, two versions of least-squares finite element methods using simple $C^0$ finite elements are developed in the paper. One is the $L^2$-LSFEM which uses linear elements, and the other is the W-LSFEM with a mesh-dependent weight to ensure the optimal convergence. Under the assumption that the PDE has a unique solution, a priori and a posteriori error estimates are presented. With an extra assumption on the operator regularity, convergence in standard norms for the W-LSFEM is also discussed. $L^2$-error estimates are derived for both formulations. Extensive numerical experiments for continuous, discontinuous, and even degenerate coefficients on smooth and singular solutions are performed to test the accuracy and efficiency of the proposed methods.