Superconvergence and extrapolation of non-conforming low order finite elements applied to the Poisson equation |
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Authors: | Lin Qun; Tobiska Lutz; Zhou Aihui |
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Institution: |
1 Institute of Computational Mathematics and Scientific/Engineering Computing, Academy of Mathematics and System Sciences, Chinese Academy of Sciences, PO Box 2719, Beijing, 100080, China, 2 Institut für Analysis und Numerik, Otto-von-Guericke-Universität Magdeburg, PSF 4120, D-39016 Magdeburg, Germany, 3 Institute of Computational Mathematics and Scientific/Engineering Computing, Academy of Mathematics and System Sciences, Chinese Academy of Sciences, PO Box 2719, Beijing, 100080, China
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Abstract: | It is well-known that on uniform meshes the piecewise linearconforming finite element solution of the Poisson equation approximatesthe interpolant to a higher order than the solution itself.In this paper, this type of superclose property is studied forthe canonical interpolant defined by the nodal functionals ofseveral non-conforming finite elements of lowest order. By givingexplicit examples we show that some non-conforming finite elementsdo not admit the superclose property. In particular, we discusstwo non-conforming finite elements which satisfy the supercloseproperty. Moreover, applying a postprocessing technique, wecan also state a superconvergence property for the discretizationerror of the postprocessed discrete solution to the solutionitself. Finally, we show that an extrapolation technique leadsto a further improvement of the accuracy of the finite elementsolution. |
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Keywords: | non-conforming finite elements superconvergence postprocessing extrapolation |
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