Date of Award
Doctor of Philosophy (PhD)
Thomas A. Manteuffel
Stephen F. McCormick
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 ||uh − ũ 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.
Liu, Kuo, "Hybrid First-Order System Least-Squares Finite Element Methods With The Application To Stokes And Navier-Stokes Equations" (2012). Applied Mathematics Graduate Theses & Dissertations. 30.