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Derivative superconvergence of linear finite elements by recovery techniques. (English) Zbl 1059.65096
The authors provide three formulae for recovering the value of its gradient at interior node points from a solution of an elliptic equation. These formulæ are shown to be accurate to \(O(h^2\log h)\) for the absolute value of the gradient, where \(h\) is a measure of element size. The focus of the paper is on equations in two dimensions. Quadrilateral mesh elements are assumed to be approximately parallelograms (opposite sides differ by \(O(h^2)\) when regarded as vectors) and adjacent pairs of triangular mesh elements must form approximate parallelograms.
For linear triangular elements, the derivative at an interior node \(P\) can be approximated as the arithmetic mean of derivatives in each element containing the node \(P\). For bilinear rectangular elements, an area-weighted mean of centerpoint derivative values is used, although the arithmetic mean can be used if adjacent elements vary by \(O(h^2)\) in area. For bilinear isoparametric quadrilaterals, an interpolation from gradients at midpoints of the four edges emanating from \(P\) is used.

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
35J25 Boundary value problems for second-order elliptic equations