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Sharp maximum norm error estimates for general mixed finite element approximations to second order elliptic equations. (English) Zbl 0673.65060
New families of mixed finite elements to approximate second order elliptic problems have been introduced as alternatives to the usual Raviart-Thomas Nedelec spaces. Two families, one based on simplices and the other on rectangles, were proposed by F. Brezzi, J. Douglas jun. and L. D. Marini [Numer. Math. 47, 217-235 (1985; Zbl 0599.65072)] in 2-D and by F. Brezzi, J. Douglas jun., R. Durán and M. Fortin [ibid. 51, 237-250 (1987; Zbl 0631.65107)] in 3-D.
The aim of the present paper is to derive sharp error bounds in \(L^{\infty}\) in a general framework and to apply the abstract results to obtain rates of convergence for the above families in 2-D and 3-D. The basic tool in proving the abstract results is Nitsche’s method of weighted Sobolev norms. It is also demonstrated that the abstract framework provides optimal error bounds according to approximation theory. It is also shown that the difference between the \(L^ 2\)- projection of the scalar field and the discrete solution superconverges in \(L^{\infty}\). The hybridization process is analyzed and further informations provided by the Lagrange multipliers are exploited.
Reviewer: V.Arnăutu (Iaşi)

MSC:
65N15 Error bounds for boundary value problems involving PDEs
65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
35J25 Boundary value problems for second-order elliptic equations
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