A review of two different approaches for superconvergence analysis (Q1979011)

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.