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Author: Lehrenfeld, Christoph × Has files: Yes ×

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    Locking free and gradient robust H(div)-conforming HDG methods for linear elasticity
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2020) Fu, Guosheng; Lehrenfeld, Christoph; Linke, Alexander; Streckenbach, Timo
    Robust discretization methods for (nearly-incompressible) linear elasticity are free of volume-locking and gradient-robust. While volume-locking is a well-known problem that can be dealt with in many different discretization approaches, the concept of gradient-robustness for linear elasticity is new. We discuss both aspects and propose novel Hybrid Discontinuous Galerkin (HDG) methods for linear elasticity. The starting point for these methods is a divergence-conforming discretization. As a consequence of its well-behaved Stokes limit the method is gradient-robust and free of volume-locking. To improve computational efficiency, we additionally consider discretizations with relaxed divergence-conformity and a modification which re-enables gradient-robustness, yielding a robust and quasi-optimal discretization also in the sense of HDG superconvergence.
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    Towards computable flows and robust estimates for inf-sup stable FEM applied to the time-dependent incompressible Navier-Stokes equations
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2017) Schroeder, Philipp W.; Lehrenfeld, Christoph; Linke, Alexander; Lube, Gerd
    Inf-sup stable FEM applied to time-dependent incompressible Navier-Stokes flows are considered. The focus lies on robust estimates for the kinetic and dissipation energies in a twofold sense. Firstly, pressure-robustness ensures the fulfilment of a fundamental invariance principle and velocity error estimates are not corrupted by the pressure approximability. Secondly, Re-semirobustness means that constants appearing on the right-hand side of kinetic and dissipation energy error estimates (including Gronwall constants) do not explicitly depend on the Reynolds number. Such estimates rely on the essential regularity assumption which is discussed in detail. In the sense of best practice, we review and establish pressure- and Re-semirobust estimates for pointwise divergence-free H1-conforming FEM (like Scott-Vogelius pairs or certain isogeometric based FEM) and pointwise divergence-free H(div)-conforming discontinuous Galerkin FEM. For convection-dominated problems, the latter naturally includes an upwind stabilisation for the velocity which is not gradient-based.