Revue internationale Henry Bauchau n°7 – 2015

Sommaire

1 Introduction 1
1.1 Motivation . 1
1.2 Overview 8
2 The discontinuous Galerkin method 11
2.1 Model equations11
2.2 Elements and functional spaces . 12
2.3 Variational formulation 14
2.4 Shape functions23
3 Extending the DG variational formulation 27
3.1 Stability of the interior penalty method on hybrid meshes27
3.2 Non-conformal formulation . 35
3.3 Frequential formulation of the homentropic LEE . 44
4 Iterative methods 57
4.1 Newton methods . 59
4.2 Multigrid methods 63
4.3 Concluding remarks . 71
5 Efficient data structures 75
5.1 Algebraic primitives on the computer . 76
5.2 Data Structures84
5.3 Efficient assembly . 89
5.4 Conclusions 102
6 noFUDGe: a first industrial application 105
6.1 Description of the flow 106
6.2 Computational setup . 106
6.3 Comparison of computed flow fields108
6.4 Validation . 114
6.5 Comparison of computational cost. 116
6.6 Scaling tests 116
6.7 Conclusions 118
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ii CONTENTS
7 Current status and prospects 119
7.1 Conclusions 119
7.2 Current status of the Argo group 120
7.3 Prospects 121
A Elements of functional analysis A.3
A.1 Hilbert spaces. A.3
A.2 Solvability of variational problems. A.4
A.3 The Lax-Milgram theoremA.5
A.4 The most simple exampleA.5
B Function spaces, reference elements and quadrature A.7
B.1 Construction of Lagrange interpolants . A.7
B.2 Interpolation on the boundary A.8
B.3 Specific elements . A.9
B.4 Quadrature rules . A.12
C Sharp values for the trace inverse inequality A.15
C.1 Simplices A.16
C.2 Outline ofWarburton's method . A.16
C.3 Tensor product elements. A.17
C.4 Wedges . A.19
C.5 Lagrange interpolation on pyramidsA.21
C.6 The Pascal space on the pyramid A.25
D Nonlinear instability of quadrature-free methods A.27
D.1 Original formulation . A.27
D.2 Extension to non-linear equations of state . A.30
D.3 Spurious modesA.30
Bibliography i