Date of Award

Spring 1-1-2012

Document Type

Dissertation

Degree Name

Doctor of Philosophy (PhD)

Department

Applied Mathematics

First Advisor

Thomas A. Manteuffel

Second Advisor

Stephen F. McCormick

Third Advisor

John Ruge

Fourth Advisor

Marian Brezina

Fifth Advisor

Xiao-chuan Cai

Abstract

This thesis combines the FOSLS method with the FOSLL* method to create a Hybrid method. The FOSLS approach minimizes the error, e h = uh u, 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*wh u||. 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 ||uhũ h[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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