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Computing upper bounds on Friedrichs' constant
Vejchodský, Tomáš
This contribution shows how to compute upper bounds of the optimal constant in Friedrichs’ and similar inequalities. The approach is based on the method of a prioria posteriori inequalities [9]. However, this method requires trial and test functions with continuous second derivatives. We show how to avoid this requirement and how to compute the bounds on Friedrichs’ constant using standard finite element methods. This approach is quite general and allows variable coefficients and mixed boundary conditions. We use the computed upper bound on Friedrichs’ constant in a posteriori error estimation to obtain guaranteed error bounds.
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Integration in higher-order finite element method in 3D
Kůs, Pavel
Integration of higher-order basis functions is an important issue, that is not as straightforward as it may seem. In traditional low-order FEM codes, the bulk of computational time is a solution of resulting system of linear equations. In the case of higher-order elements the situation is different. Especially in three dimensions the time of integration may represent significant part of the computation.
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Complementarity - the way towards guaranteed error estimates
Vejchodský, Tomáš
This paper presents a review of the complementary technique with the emphasis on computable and guaranteed upper bounds of the approximation error. For simplicity, the approach is described on a numerical solution of the Poisson problem. We derive the complementary error bounds, prove their fundamentals properties, present the method of hypercircle, mention possible generalizations and show a couple of numerical examples.
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