On the relationship between mixed and Galerkin finite element methods
Abstract
In the first chapter, basic error estimates are derived for the lowest-order Raviart-Thomas mixed method with the variable coefficients of second order elliptic equations projected into finite element spaces. It is shown that the method is equivalent to a nonconforming method modified by augmenting the usual $P\sb1$-nonconforming space with $P\sb2$-bubbles and that the approximate solution produced by this method can be computed from the solution of the $P\sb1$-nonconforming method modified in a virtually cost-free manner. Some new error estimates for the methods considered are obtained. In the second chapter, an abstract framework under which an equivalence between mixed finite element methods and certain modified versions of conforming and nonconforming finite element methods is established for second order elliptic problems. It is shown, based on the equivalence, that mixed methods can be implemented through usual conforming or nonconforming methods modified in a cost-free manner and that new error estimates for these methods can be derived. The Raviart-Thomas, Brezzi-Douglas-Marini, and Marini-Pietra mixed methods for second order elliptic problems are analyzed by means of the present techniques.
Degree
Ph.D.
Advisors
Douglas, Purdue University.
Subject Area
Mathematics
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