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Author: Carstensen, Carsten × Publisher: Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik ×

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Now showing 1 - 3 of 3
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    Guaranteed error control for the pseudostress approximation of the Stokes equations
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2015) Bringmann, Philipp; Carstensen, Carsten; Merdon, Christian
    The pseudostress approximation of the Stokes equations rewrites the stationary Stokes equations with pure (but possibly inhomogeneous) Dirichlet boundary conditions as another (equivalent) mixed scheme based on a stress in H (div) and the velocity in L2. Any standard mixed finite element function space can be utilized for this mixed formulation, e.g. the Raviart-Thomas discretization which is related to the Crouzeix-Raviart nonconforming finite element scheme in the lowest-order case. The effective and guaranteed a posteriori error control for this nonconforming velocity-oriented discretization can be generalized to the error control of some piecewise quadratic velocity approximation that is related to the discrete pseudostress. The analysis allows for local inf-sup constants which can be chosen in a global partition to improve the estimation. Numerical examples provide strong evidence for an effective and guaranteed error control with very small overestimation factors even for domains with large anisotropy.
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    Reliable averaging for the primal variable in the Courant FEM and hierarchical error estimators on red-refined meshes
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2016) Carstensen, Carsten; Eigel, Martin
    A hierarchical a posteriori error estimator for the first-order finite element method (FEM) on a red-refined triangular mesh is presented for the 2D Poisson model problem. Reliability and efficiency with some explicit constant is proved for triangulations with inner angles smaller than or equal to π/2 . The error estimator does not rely on any saturation assumption and is valid even in the pre-asymptotic regime on arbitrarily coarse meshes. The evaluation of the estimator is a simple post-processing of the piecewise linear FEM without any extra solve plus a higher-order approximation term. The results also allows the striking observation that arbitrary local averaging of the primal variable leads to a reliable and efficient error estimation. Several numerical experiments illustrate the performance of the proposed a posteriori error estimator for computational benchmarks.
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    Aspects of quaranteed error control in CPDEs
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2013) Carstensen, Carsten; Merdon, Christian; Neumann, Johannes
    Whenever numerical algorithms are employed for a reliable computational forecast, they need to allow for an error control in the final quantity of interest. The discretisation error control is of some particular importance in computational PDEs (CPDEs) where guaranteed upper error bounds (GUB) are of vital relevance. After a quick overview over energy norm error control in second-order elliptic PDEs, this paper focuses on three particular aspects. First, the variational crimes from a nonconforming finite element discretisation and guaranteed error bounds in the discrete norm with improved postprocessing of the GUB. Second, the reliable approximation of the discretisation error on curved boundaries and, finally, the reliable bounds of the error with respect to some goal-functional, namely, the error in the approximation of the directional derivative at a given point