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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 ThomĂ©e 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.

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