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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.

##### 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 35J25 Boundary value problems for second-order elliptic equations