Hybrid-FOSLS with Application to Stokes and Navier-Stokes Equations
Kuo Liu
Applied Mathematics, University of Colorado Boulder
Date and time:
Tuesday, September 18, 2012 - 3:30pm
Abstract:
Hybrid-FOSLS is based on combining the FOSLS method with the FOSLL* method. The FOSLS approach minimizes the error, eh = uh - u , over a finite element subspace, Vh, in the operator norm, minuh ∈ Vh|L(uh-u|. The FOSLL* method looks for an approximation in the range of L*, setting uh=L*wh and choosing wh ∈ Wh, a standard finite element space. FOSLL* minimizes the L2 norm of the error over L*(Wh), that is, minwh ∈ Wh |L*wh - u|. FOSLS enjoys a locally sharp, globally reliable, and easily computable a posteriorierror 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, uh, that is nearly the optimal over Vh in the graph norm, |eh|2G := ½|eh|2 + |Leh|2. The FOSLS and intermediate terms in the Hybrid functional provide a very effective a posteriori error measure.
In this talk we show that the hybrid functional is coercive and continuous in graph-like norm with modest coercivity and continuity constants, c0 = 1/3 and c1 = 3; that both |eh| and |Leh| converge with rates based on standard interpolation bounds; and that, if LL* has full H2-regularity, the L2 error, |eh|, converges with a full power of the discretization parameter, h, faster than the functional norm. Letting ũh denote the optimum over Vh in the graph norm, we also show that if superposition is used, then |uh-ũh|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.