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Uniform superconvergence estimates of derivatives for the finite element method. (Chinese. English summary) Zbl 0549.65073
Summary: Suppose that on a C-uniform (trilateral or quadrilateral) partition, $$S^ h\subset\overset \circ W^ 1_{\infty}(\Omega)$$ consists of a given class of piecewise polynomials, whose degree is not greater than k. Let $$u^ h\in S^ h$$ be a finite element solution of $$u\in W_{\infty}^{k+2}(\Omega)$$ and let $$u^ 1\in S^ h$$ be an interpolation of u. In this paper we obtain the following superconvergence estimates: $$(1)\quad| u^ h-u^ I|_{1,\infty,\Omega}\leq C|\ln h|^{\mu}h^{k+1}\| u\|_{k+2,\infty,\Omega}$$ where $$\mu =const\geq 1$$. We have, further, uniform suerconvergene estimates $$(2)\quad| \bar D_ l(u-u^ h)(Q)|\leq c|\ln h|^{\mu}h^{k+1}\| u\|_{k+2,\infty,\Omega},\quad (k\geq 1)$$ for all directions l on all optimal points Q. At present, estimates (1), (2) are optimal results about uniform superconvergence for derivatives, in which the convergence rates for derivatives $$(k=1)$$ increase in accordance with square. Abroad, not only Zlamal who hasn’t obtained this, even Bramble and Schatz and Thomée have obtained higher order local accuracy only for higher degree elements by averaging on a so-called stronger ”uniform mesh” on $$\Omega_ 0\subset\subset \Omega_ 1\subset\subset \Omega$$. However, the practical calculus of finite elements can not be used at present.

##### 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