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Hybrid-FOSLS with Application to Stokes and Navier-Stokes Equations

Kuo Liu

Ìý

Applied Mathematics,Ìý

Date and time:Ìý

Tuesday, September 18, 2012 - 3:30pm

Abstract:Ìý

Hybrid-FOSLS is based on combining the FOSLS method with the FOSLL* method. The FOSLS approach minimizes the error,ÌýehÌý=ÌýuhÌý-ÌýuÌý, over a finite element subspace, Vh, in the operator norm, minuhÌý∈ Vh|L(uh-u|. The FOSLL* method looks for an approximation in the range ofÌýL*, settingÌýuh=L*whÌýand choosingÌýwhÌý∈ÌýWh, a standard finite element space. FOSLL* minimizes theÌýL2Ìýnorm of the error overÌýL*(Wh), that is, minwhÌý∈ÌýWhÌý|L*whÌý-Ìýu|. FOSLS enjoys a locally sharp, globally reliable, and easily computableÌýa 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,Ìýuh, that is nearly the optimal over VhÌýin the graph norm, |eh|2GÌý:= ½|eh|2Ìý+ |Leh|2. The FOSLS and intermediate terms in the Hybrid functional provide a very effectiveÌýa posterioriÌýerror measure.

In this talk we show that the hybrid functional is coercive and continuous in graph-like norm with modest coercivity and continuity constants, c0Ìý= 1/3 and c1Ìý= 3; that both |eh| and |Leh| converge with rates based on standard interpolation bounds; and that, if LL* has full H2-regularity, the L2Ìýerror, |eh|, converges with a full power of the discretization parameter,Ìýh, faster than the functional norm. Letting Å©hÌýdenote the optimum over VhÌýin the graph norm, we also show that if superposition is used, then |uh-Å©h|GÌýconverges two powers ofÌýhÌýfaster than the functional norm. Numerical tests are provided to confirm the efficiency of the Hybrid method and effectiveness of theÌýa posterioriÌýerror measure.