A review of two different approaches for superconvergence analysis. (English) Zbl 0938.65128

The paper surveys several methods leading to superconvergence phenomena when solving linear second-order elliptic boundary value problems by the finite element method. These methods include Green’s function method, tensor convolution methods, averaging convolution methods, energy-orthogonalization methods, interpolate postprocessing methods and asymptotic expansion methods. The author compares theoretical results obtained in China and “Western” countries. The Chinese approach enables us to prove superconvergence up to the boundary of a domain in question for a smooth solution. Triangulations used can be, e.g., quasi-uniform, which means that any two adjacent triangles form an approximate parallelogram. In this case triangulations are not locally symmetric with respect to a point. Superconvergence phenomena can also be obtained for piecewise quasi-uniform meshes, which is important from a practical point of view.


65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
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