Hybrid First-Order System Least Squares Finite Element Methods with Application to Stokes Equations

Abstract

This paper combines first-order system least squares (FOSLS) with first-order system $LL^*$ (FOSLL$^*$) to create a Hybrid method. The FOSLS approach minimizes the error, ${\bf e}^h = {\bf u}^h - {\bf u}$, over a finite element subspace, ${\cal V}^h$, in the operator norm: $\min_{{\bf u}^h\in{\cal V}^h}\| L ({\bf u}^h-{\bf u})\|$. The FOSLL$^*$ method looks for an approximation in the range of $L^*$, setting ${\bf u}^h = L^*{\bf w}^h$ and choosing ${\bf w}^h \in {\cal W}^h$, a standard finite element space. FOSLL$^*$ minimizes the $L^2$ norm of the error over $L^*({\cal W}^h)$: $\min_{{\bf w}^h\in{\cal W}^h} \| L^*{\bf w}^h - {\bf u}\|$. FOSLS enjoys a locally sharp, globally reliable, and easily computable a posteriori error estimate, while FOSLL$^*$ does not. However, FOSLL$^*$ has the major advantage that it applies to problems that do not exhibit enough smoothness to enable the full advantages that the FOSLS approach otherwise provides. The Hybrid method attempts to retain the best properties of both FOSLS and FOSLL$^*$. This is accomplished by combining the FOSLS functional, the FOSLL$^*$ functional, and an intermediate term that draws them together. The Hybrid method produces an approximation, ${\bf u}^h$, that is nearly the optimal over ${\cal V}^h$ in the graph norm, $\|{\bf e}^h\|_{{\cal G}}^2:= \frac{1}{2}\|{\bf e}^h\|^2 + \|L{\bf e}^h\|^2$. The FOSLS and intermediate terms in the Hybrid functional provide a very effective a posteriori error measure. This paper establishes that the Hybrid functional is coercive and continuous in a graph-like norm with modest coercivity and continuity constants, $c_0 = 1/3$ and $c_1=3$; that both $\|{\bf e}^h \|$ and $\|L {\bf e}^h\|$ converge with rates based on standard interpolation bounds; and that if $LL^*$ has full $H^2$ regularity, the $L^2$ error, $\|{\bf e}^h\|$, converges with a full power of the discretization parameter, $h$, faster than the functional norm. Letting $\tilde{{\bf u}}^h$ denote the optimum over ${\cal V}^h$ in the graph norm, this paper also shows that if superposition is used with nested iteration, then $\| {\bf u}^h-\tilde{{\bf u}}^h\|_{{\cal G}}$ converges two powers of $h$ faster than the functional norm. Numerical tests are provided to confirm the efficiency of the Hybrid method and effectiveness of the a posteriori error measure.