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Superconvergence for the gradient of finite element approximations by L$$^{2}$$ projections. (English) Zbl 1047.65095
The authors propose and analyze a gradient recovery technique for finite element solutions which provides new gradient approximations with high order of accuracy. They modify the method of O. C. Zienkiewicz and J. Z. Zhu [Comput. Methods Appl. Mech. Eng. 101, No. 1–3, 207–224 (1992; Zbl 0779.73078)] by applying a global least-squares fitting to the gradient of finite element approximation and provide a theoretical analysis for the modified Zienkiewicz-Zhu method by establishing a superconvergence estimate for the recovered gradient/flux on general quasi-uniform meshes. They also give numerical results to show that the recovery technique is robust and efficient.

##### MSC:
 65N12 Stability and convergence of numerical methods for boundary value problems involving PDEs 65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs 65N15 Error bounds for boundary value problems involving PDEs 65N50 Mesh generation, refinement, and adaptive methods for boundary value problems involving PDEs 35J25 Boundary value problems for second-order elliptic equations
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