Graduate Thesis Or Dissertation

 

Hybrid First-Order System Least-Squares Finite Element Methods With The Application To Stokes And Navier-Stokes Equations Public Deposited

https://scholar.colorado.edu/concern/graduate_thesis_or_dissertations/cr56n0995
Abstract
  • This thesis combines the FOSLS method with the FOSLL* method to create a Hybrid method. The FOSLS approach minimizes the error, e ʰ = uʰu, over a finite element subspace, [special characters omitted], in the operator norm, [special characters omitted] ||L(uʰ − u)||. The FOSLL* method looks for an approximation in the range of L*, setting uʰ = L*wʰ and choosing wʰ ∈ [special characters omitted], a standard finite element space. FOSLL* minimizes the L ² norm of the error over L*([special characters omitted]), that is, [special characters omitted] ||L*wʰu||. 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, uʰ, that is nearly the optimal over [special characters omitted] in the graph norm, ||eʰ[special characters omitted] := ½||eʰ|| ² + ||Leʰ|| ². The FOSLS and intermediate terms in the Hybrid functional provide a very effective a posteriori error measure.

    In this dissertation 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 ||L eʰ|| 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 [special characters omitted] in the graph norm, we also show that if superposition is used, then ||uʰũʰ||[special characters omitted] converges two powers of h faster than the functional norm. Numerical tests on are provided to confirm the efficiency of the Hybrid method and effectiveness of the a posteriori error measure.

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  • 2012
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  • 2020-01-13
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