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On \(L^{\infty}\)-accuracy of mixed finite element methods for second order elliptic problems. (English) Zbl 0677.65103
The authors are interested in the Dirichlet problem on a bounded domain for the equation \(Lu=-\nabla \cdot (a(x)\nabla u+b(x)u+c(x)u=f.\) This equation is expressed in a mixed formulation (with \(p=-a\nabla u+bu)\) as \((a^{-1}p,q)-(\nabla q,u)+(ba^{-1}u,q)=0,\) \(\forall q\), \((\nabla \cdot p,v)+(cu,v)=(f,v)\) \(\forall v\), where q and v are in appropriate spaces and the \(L^ 2\) inner product is used. The authors are able to prove \(L^{\infty}\) error estimates for the solution of the mixed problem using Raviart-Thomas-Nedelec finite elements of order k over either triangles or parallelograms. They also demonstrate superconvergence at selected points for parallelograms and for order \(k=0\) over triangles.
In particular, the authors are able to show, for \(k\geq 1\), that \(\| u- u_ h\|_{L^{\infty}}+\| p-p_ h\|_{L^{\infty}}\leq Ch^{k+1}\| u\|_{W^{k+2,\infty}}\) and for \(k=0\) that \(\| u- u_ h\|_{L^{\infty}}+\| p-p_ h\|_{L^{\infty}}\leq ch(c_ f+\| u\|_{W^{2,\infty}}).\) The authors are also able to show that for parallelogram elements superconvergence is exhibited on the set S of Gauß-Legendre points: \(\max \{| u(x)-u_ h(x)|:\quad x\in S\}=O(h^{k+2}| \log h|^ 2),\) and for triangular elements superconvergence is exhibited solely for elements of order \(k=0\) at the set \(\hat S\) of barycenters: \(\max \{| u(x)-u_ h(x)|:\quad x\in \hat S\}=O(h^ 2| \log h|^ 2).\)
Reviewer: Myron Sussman

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