Abstract
We study the problem of Lagrange interpolation of functions of two variables by quadratic polynomials under the condition that nodes of interpolation are vertices of a triangulation. For an extensive class of triangulations we prove that every inner vertex belongs to a local six-tuple of vertices which, used as nodes of interpolation, have the following property: For every smooth function there exists a unique quadratic Lagrange interpolation polynomial and the related local interpolation error is of optimal order. The existence of such six-tuples of vertices is a precondition for a successful application of certain post-processing procedures to the finite-element approximations of the solutions of differential problems.
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This work was supported by the grant GA ČR 103/05/0292.
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Dalík, J. Optimal-order quadratic interpolation in vertices of unstructured triangulations. Appl Math 53, 547–560 (2008). https://doi.org/10.1007/s10492-008-0041-x
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DOI: https://doi.org/10.1007/s10492-008-0041-x