Browsing by Author "Carstensen, Carsten"

Now showing 1 - 11 of 11
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    Advanced Computational Engineering
    (Zürich : EMS Publ. House, 2012) Carstensen, Carsten; Schröder, Jörg; Wriggers, Peter
    The finite element method is the established simulation tool for the numerical solution of partial differential equations in many engineering problems with many mathematical developments such as mixed finite element methods (FEMs) and other nonstandard FEMs like least-squares, nonconforming, and discontinuous Galerkin (dG) FEMs. Various aspects on this plus related topics ranging from order-reduction methods to isogeometric analysis has been discussed amongst the pariticpants form mathematics and engineering for a large range of applications.
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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
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    Computational Engineering
    (Zürich : EMS Publ. House, 2015) Carstensen, Carsten; Demkowicz, Leszek; Wriggers, Peter
    The focus of this Computational Engineering Workshop was on the mathematical foundation of state-of-the-art and emerging finite element methods in engineering analysis. The 52 participants included mathematicians and engineers with shared interest on discontinuous Galerkin or Petrov-Galerkin methods and other generalized nonconforming or mixed finite element methods.
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    Computational Engineering
    (Zürich : EMS Publ. House, 2018) Buffa, Annalisa; Carstensen, Carsten; Schroeder, Joerg
    This Workshop treated a variety of finite element methods and applications in computational engineering and expanded their mathematical foundation in engineering analysis. Among the 53 participants were mathematicians and engineers with focus on mixed and nonstandard finite element schemes and their applications.
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    Gemischte und nicht-standard Finite-Elemente-Methoden mit Anwendungen
    (Zürich : EMS Publ. House, 2005) Carstensen, Carsten; Hackl, Klaus
    Mixed and non-conforming finite element methods form a general mathematical framework for the spatial discretisation of partial differential equations, mainly applied to elliptic equations of second order and are becoming increasingly important for the solution of nonlinear problems. These methods are under active discussion in the mathematical and the engineering community and aim of the workshop was to provide a joint forum for the current state of research.
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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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    Mini-Workshop: Convergence of Adaptive Algorithms
    (Zürich : EMS Publ. House, 2005) Carstensen, Carsten; Dörfler, Willy
    Adaptive refinement strategies are a key concept for the efficient numerical solution of partial differential equations. New findings allow the study of convergence and optimality of adaptive finite element methods. It was the aim of the Oberwolfach to discuss necessary conditions for optimal convergence and to extend the theory to new areas of application.
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    Mixed Finite Element Methods and Applications
    (Oberwolfach-Walke : Mathematisches Forschungsinstitut Oberwolfach, 2001) Carstensen, Carsten; Hoppe, Ronald H.W.
    [no abstract available]
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    Nonstandard Finite Element Methods
    (Zürich : EMS Publ. House, 2008) Carstensen, Carsten; Monk, Peter
    [no abstract available]
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    Nonstandard Finite Element Methods (hybrid meeting)
    (Zürich : EMS Publ. House, 2021) Carstensen, Carsten; Ern, Alexandre; Hu, Jun
    Finite element methodologies dominate the computational approaches for the solution to partial differential equations and nonstandard finite element schemes most urgently require mathematical insight in their design. The hybrid workshop vividly enlightened and discussed innovative nonconforming and polyhedral methods, discrete complex-based finite element methods for tensor-problems, fast solvers and adaptivity, as well as applications to challenging ill-posed and nonlinear problems.
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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.