===affil2:
Dept. of Applied Mathematics, University of Colorado at Boulder
===firstname:
Kuo
===firstname4:
John
===firstname3:
Stephen
===lastname2:
Manteuffel
===lastname:
Liu
===firstname5:
Lei
===affil6:

===lastname3:
McCormick
===email:
kuol@colorado.edu
===lastname6:

===affil5:
Dept. of Applied Mathematics, University of Colorado at Boulder
===otherauths:

===lastname4:
Ruge
===affil4:
Dept. of Applied Mathematics, University of Colorado at Boulder
===lastname7:

===affil7:

===firstname7:

===postal:
1320 Grandview Ave.,
Boulder, CO 80302
===firstname6:

===ABSTRACT:
In this talk, we combine the FOSLS method with the FOSLL$^*$ method to create a Hybrid method.
The FOSLS approach minimizes the error, $\bfe^h = \bu^h - \bu$, over a finite element
subspace, $\CV^h$,
in the operator norm,  $\min_{\bu^h\in\CV^h}\| L (\bu^h-\bu) \|$. The FOSLL$^*$ method
looks for an approximation
in the range of $L^*$, setting $\bu^h = L^*\bw^h$ and choosing $\bw^h \in \CW^h$, a
standard finite element space.
FOSLL$^*$ minimizes the $\BL^2$ norm of the error over $L^*(\CW^h)$, that is,
$\min_{\bw^h\in\CW^h} \| L^*\bw^h - \bu\|$.
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,
$\bu^h$, that is nearly the optimal over $\CV^h$ in the graph norm,
$\|\bfe^h\|_{\CG}^2:= \frac{1}{2}\|\bfe^h\|^2 + \|L\bfe^h\|^2$.
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 $\|\bfe^h \|$ and $\|L
\bfe^h\|$
converge with rates based on standard interpolation bounds; and that if  $LL^*$ has full
$H^2$ regularity,
the $\BL^2$ error, $\|\bfe^h\|$,  converges with a full power of the discretization
parameter, $h$, faster
than the functional norm. Letting $\tilde{\bu}^h$ denote the optimum over $\CV^h$ in the
graph norm, we also
show that if superposition is used, then $\| \bu^h -\tilde{\bu}^h\|_{\CG}$
converges two powers of $h$ faster than the functional norm. Numerical tests are provided
to confirm the efficiency of the Hybrid method.

===affil3:
Dept. of Applied Mathematics, University of Colorado at Boulder
===lastname5:
Tang
===affilother:

===title:
Hybrid First-order System Least Squares Finite Element Methods With Application to Stokes Equations
===firstname2:
Thomas