Error Analysis for Constrained First-Order System Least-Squares Finite-Element Methods

Abstract

In this paper, a general error analysis is provided for finite-element discretizations of partial differential equations in a saddle-point form with divergence constraint. In particular, this extends upon the work of [J. H. Adler and P. S. Vassilevski, Springer Proc. Math. Statist. 45, Springer, New York, 2013, pp. 1--19], giving a general error estimate for finite-element problems augmented with a divergence constraint and showing that these estimates are obtained for problems such as diffusion and Stokes' using the first-order system least-squares (FOSLS) finite-element method. The main result is that by enforcing the constraint on a ${\mathbf H}^1$-equivalent FOSLS formulation one maintains optimal convergence of the FOSLS functional (i.e., the energy norm of the error) while guaranteeing the conservation of the divergence constraint (i.e., mass conservation in some examples). The error estimates and results depend on using finite elements for the constraint space that are inf-sup stable when paired with the spaces used for the original unknowns. This includes using discontinuous spaces on coarse meshes and pairing with standard bilinear or biquadratic elements in order to confirm the results.