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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

##### MSC:
 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