Kuo Liu
Hybrid First-order System Least Squares Finite Element Methods With Application to Stokes Equations

1320 Grandview Ave
Boulder
CO 80302
kuol@colorado.edu
Thomas Manteuffel
Stephen McCormick
John Ruge
Lei Tang

In this talk, we combine the FOSLS method with the FOSLL$^*$ method 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, $\min_{{\bf u}^h\in{\cal V}^h}\Vert L ({\bf u}^h-{\bf u}) \Vert$ . 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 ${\bf L}^2$ norm of the error over $L^*({\cal W}^h)$ , that is, $\min_{{\bf w}^h\in{\cal W}^h} \Vert L^*{\bf w}^h - {\bf u}\Vert$ . FOSLS enjoys a locally sharp, globally reliable, and easily computable a posterior error estimate, while FOSLL$^*$ does not.

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, $\Vert{\bf e}^h\Vert _{{\cal G}}^2:= \frac{1}{2}\Vert{\bf e}^h\Vert^2 + \Vert L{\bf e}^h\Vert^2$ . The FOSLS and intermediate terms in the Hybrid functional provide a very effective a posteriori error measure.

We show that the hybrid functional is coercive and continuous in the graph-like norm with modest constants, $c_0 = 1/3$ and $c_1=3$ ; that both $\Vert{\bf e}^h \Vert$ and $\Vert L {\bf e}^h\Vert$ converge with rates based on standard interpolation bounds; and that if $LL^*$ has full $H^2$ regularity, the ${\bf L}^2$ error, $\Vert{\bf e}^h \Vert$ , 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, we also show that if superposition is used, then $\Vert {\bf u}^h -\tilde{{\bf u}}^h\Vert _{{\cal G}}$ converges two powers of $h$ faster than the functional norm. Numerical tests are provided to confirm the efficiency of the Hybrid method.