Finite volume element methods for non-definite problems

The error estimates for nite volume element method applied to and D non de nite problems are derived A simple upwind scheme is proven to be unconditionally stable and rst order accurate Mathematics Subject Classi cation N Introduction The purpose of this note is three fold We would like to extend the results due to Bank and Rose Hackbusch Cai and McCormick and Jianguo and Shitong to D problems provide a theory for non de nite equations and nally give a more exible way to obtain a priori estimates that in some sense have the avor of the rst Fix lemma in the nite element theory and generalize the technique used by Cai to analyze the e ects of numerical integration We will demonstrate this approach on a simple upwind scheme although the technique can handle more sophisticated upwind strategies see for example We consider the following boundary value problem r A x ru b x u c x u f x in a u x on b where is a open subset of R d or We refer for the extensive discus sion of solvability of the problem to the monograph by Ladyzhenskaya and Ural tseva Our approach is based on the generalization of Lax Milgram lemma due to Ne cas and modi ed by Babu ska and Aziz First we introduce some notations Let U and V be two real Hilbert spaces equipped with the norms k kU and k kV respectively and let A U V R be a bilinear form We de ne the following variational problem The author was partially supported by funding from DOE grant DE FG ER and ARO grant DAAH Find an element u U such that A u v f v v V Theorem Babu ska and Aziz Assume that there exist a positive con stants C and such that the bilinear form A U V R satis es jA u v j CkukUkvkV u U v V a sup v V v jA u v j kvkV kukU u U b sup u U jA u v j v V v c and that f V R is a continuous linear form Then the variational problem has one and only one solution and the following stability estimate holds kukU kfkV We use the standard notation for Sobolev spaces Let U V H V H let the bilinear form A be de ned by A u v A u v A u v A u v a A u v Z Aru rv dx b