Discontinuous Galerkin Methods for Viscous Incompressible Flow

Beschreibung

Guido Kanschat reviews several discontinuous Galerkin schemes for elliptic and viscous flow problems. Setting out from Nitsche's method for weak boundary conditions, he studies the interior penalty and LDG methods. Combined with a stable advection discretization, they yield stable DG methods for linear flow problems of Stokes and Oseen type which are applied to the Navier-Stokes problem. The author not only presents the analytical techniques used to study these methods but also devotes a major discussion to the efficient numerical solution of discrete problems.
the so called standard superconvergent div 6 PREFACE Contents 1 Basics 15 1.1 Meshes and shape functions ... ... .... ... .... .... ... ... 15 1.2 Spaces and approximation .... ... .... ... .... .... ... ... 19 2 Linear Diffusion I 33 3 Linear Diffusion II 67 4 Stokes Equations 89 8 CONTENTS 5 Flow Problems 105 5.1 Advection-Diffusion-ReactionEquation .. .... .... ... .... ... . 107 5.2 Oseen Equations .... ... .... ... .... .... ... .... ... . 110 5.3 Navier-Stokes Equations ... .... ... .... .... ... .... ... . 112 6 Linear Solvers 131 6.1 Krylov space methods . . . . . .... ... .... .... ... .... ... . 132 6.2 Interior Penalty . .... ... .... ... .... .... ... .... ... . 134 6.3 Local multigrid . .... ... .... ... .... .... ... .... ... . 145 6.4 LDG .... ... .... ... .... ... .... .... ... .... ... . 151 6.5 Advection Diffusion .. ... .... ... .... .... ... .... ... . 158 6.6 Stokes ... ... .... ... .... ... .... .... ... .... ... . 160 6.7 Oseen equations . .... ... .... ... ........ ... .... ... . 162 A Example problems 167 A.1 Meshes and domains .. ... .... ... .... .... ... .... ... . 167 A.2 Poisson equation .... ... .... ... .... .... ... .... ... . 170 A.3 Advection-diffusion-reaction . .... ... .... .... ... .... ... . 171 A.4 Stokes and Oseen equations .. .... ... .... .... ... .... ... . 172 List of Figures 1.1 The set and (if ). . ... .... ... .... .... ... ... 17 1.2 The cel (dark grey) and the neighbor set (grey). Note that a common corner does not result in neighborhood. . . . . .... ... .... .... ... ... 19 2.1 Smallest eigenvalue of on a mesh of squares . .... .... ... ... 38 2.2 Eigenfunction to the smallest eigenvalue of for (left) and (right) on a mesh of squares . . . . . .... ... .... .... ... ... 39 2.3 Accuracy of the interior penalty method depending on the parameter . . ... 44 2.4 Pointwise errors on irregular meshes . . .... ... .... .... ... ... 55 2.5 Adapted mesh (level ) for the point value computation .... ... ... 60 2.6 -error over number of degrees of freedom for CG- , DG- and - elements (left) and (bi-)quadratic elements (right) .. .... .... ... ... 62
Guido Kanschat reviews several discontinuous Galerkin schemes for elliptic and viscous flow problems. Setting out from Nitsche's method for weak boundary conditions, he studies the interior penalty and LDG methods. Combined with a stable advection discretization, they yield stable DG methods for linear flow problems of Stokes and Oseen type which are applied to the Navier-Stokes problem. The author not only presents the analytical techniques used to study these methods but also devotes a major discussion to the efficient numerical solution of discrete problems.

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Guido Kanschat

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