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Error estimation and adaptive mesh refinement for aerodynamic flows

By Ralf Hartmann and Paul Houston

Abstract

This lecture course covers the theory of so-called duality-based a posteriori error estimation of DG finite element methods. In particular, we formulate consistent and adjoint consistent DG methods for the numerical approximation of both the compressible Euler and Navier-Stokes equations; in the latter case, the viscous terms are discretized based on employing an interior penalty method. By exploiting a duality argument, adjoint-based a posteriori error indicators will be established. Moreover, application of these computable bounds within automatic adaptive finite element algorithms will be developed. Here, a variety of isotropic and anisotropic adaptive strategies, as well as $hp$-mesh refinement will be investigated

Publisher: von Karman Institute for Fluid Dynamics
OAI identifier: oai:eprints.nottingham.ac.uk:1193
Provided by: Nottingham ePrints

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  1. (1999). A discontinuous hp finite element method for the Euler and NavierStokes equations. doi
  2. (1998). A discontinuous hp-finite element method for diffusion problems.
  3. (1996). A feed-back approach to error control in finite element methods: Basic analysis and examples. doi
  4. (2002). A fully automatic hp-adaptivity.
  5. (1997). A high-order accurate discontinuous finite element method for the numerical solution of the compressible Navier-Stokes equations. doi
  6. (2000). A multilinear singular value decomposition.
  7. (2006). A note on the design of hp–version interior penalty discontinuous Galerkin finite element methods for degenerate problems.
  8. (1995). A parallel adaptive strategy for hp finite element computations.
  9. (1999). A posteriori error analysis for numerical approximations of Friedrichs systems.
  10. (1483). A posteriori error analysis for stabilised finite element approximations of transport problems.
  11. (1999). A posteriori error estimation for anisotropic tetrahedral and triangular finite element meshes.
  12. (1994). A selection of experimental test cases for the validation of CFD codes,
  13. (2007). A triangular cut-cell adaptive method for high-order discretizations of the compressible Navier-Stokes equations.
  14. (2002). Adaptive discontinuous Galerkin finite element methods for nonlinear hyperbolic conservation laws. doi
  15. (2002). Adaptive discontinuous Galerkin finite element methods for the compressible Euler equations. doi
  16. (2006). Adaptive discontinuous Galerkin methods with shock-capturing for the compressible Navier-Stokes equations. doi
  17. (2000). Adaptive FE Methods for Conservation Equations.
  18. (2002). Adaptive finite element approximation of hyperbolic problems.
  19. (2002). Adaptive Finite Element Methods for the Compressible Euler Equations. doi
  20. (1994). Adaptive mesh strategies for the spectral element method.
  21. (2006). Adaptivity and a posteriori error estimation for DG methods on anisotropic meshes.
  22. (2009). ADGIMA – A European project on the development of adaptive higher-order variational methods for aerospace applications.
  23. (2007). Adjoint consistency analysis of discontinuous Galerkin discretizations. doi
  24. (1997). Adjoint equations in CFD: duality, boundary conditions and solution behaviour.
  25. (1998). An adaptive refinement strategy for hp–finite element computations. doi
  26. (2007). An adaptive strategy for hp-FEM based on testing for analyticity.
  27. (2000). An adaptive-order discontinuous Galerkin method for the solution of the Euler equations of gas dynamics. doi
  28. (2001). An error indicator for mortar element solutions to the Stokes problem.
  29. (2000). An h-p adaptive spectral element method for Stokes flow. doi
  30. (2004). An H1(P h)-coercive discontinuous Galerkin formulation for the Poisson problem: 1-d analysis. Master’s thesis, doi
  31. An hp-adaptive mixed discontinuous Galerkin FEM for nearly incompressible linear elasticity.
  32. (1993). An hp-adaptive strategy.
  33. (1997). An hp–adaptive discontinuous Galerkin FEM for computational fluid dynamics.
  34. (2001). An optimal control approach to a posteriori error estimation in finite element methods.
  35. (2008). An optimal order interior penalty discontinuous Galerkin discretization of the compressible Navier–Stokes equations.
  36. (2007). Anisotropic Adaptive Refinement For Discontinuous Galerkin Methods.
  37. (1999). Anisotropic finite elements: Local estimates and applications. doi
  38. (2006). Anisotropic mesh refinement for discontinuous Galerkin methods in aerodynamic flow simulations. Diploma thesis,
  39. (2008). Anisotropic mesh refinement for discontinuous Galerkin methods in two-dimensional aerodynamic flow simulations.
  40. (1997). Anisotropic unstructured mesh adaption for flow simulations.
  41. (2003). Applications of Lobatto polynomials to an adaptive finite element method: A posteriori error estimates for hp–adaptivity and grid-to-grid interpolation.
  42. (1998). Computational methods for singularly perturbed systems. In Analyzing multiscale phenomena using singular perturbation methods
  43. (2007). deal.II – A general purpose object oriented finite element library. doi
  44. (2009). deal.II Differential Equations Analysis Library, Technical Reference.
  45. (1997). Diffusion and Elliptic Operators.
  46. (2007). Discontinuous Galerkin methods for advection–diffusion– reaction problems on anisotropically refined meshes.
  47. (2009). Discontinuous Galerkin methods for computational aerodynamics – 3D adaptive flow simulation with the DLR PADGE code. Aerospace Science and Technology, doi
  48. (2007). Discontinuous Galerkin methods on hp–anisotropic meshes I: A priori error analysis.
  49. (2009). Discontinuous Galerkin methods on hp–anisotropic meshes II: A posteriori error analysis and adaptivity.
  50. (2003). Discontinuous Galerkin methods on shape-regular and anisotropic meshes.
  51. (2005). Discontinuous Galerkin solution of the Reynoldsaveraged Navier-Stokes and k−ω turbulence model equations. doi
  52. (2002). Discontinuous hp-finite element methods for advection– diffusion–reaction problems.
  53. (2003). Duality-based adaptivity in the hp-finite element method.
  54. (2009). Error estimation and anisotropic mesh refinement for 3d aerodynamic flow simulations.
  55. (1985). Exact solutions to transonic and supersonic flows.
  56. (2000). GMRES discontinuous Galerkin solution of the compressible NavierStokes equations.
  57. (2003). Goal-oriented a posteriori error estimation for multiple target functionals. In
  58. (2004). Goal-oriented hp-adaptivity for elliptic problems.
  59. (1997). High-order accurate discontinuous finite element solution of the 2d Euler equations. doi
  60. (1995). hp-version discontinuous Galerkin methods for hyperbolic conservation laws: a parallel adaptive strategy.
  61. (2001). hp–Adaptive discontinuous Galerkin finite element methods for hyperbolic problems.
  62. (2001). hp–Adaptive two–level methods for boundary integral equations on curves.
  63. (2000). hp–DGFEM for partial differential equations with nonnegative characteristic form.
  64. (2002). hp–Discontinuous Galerkin finite element methods for nonlinear hyperbolic problems.
  65. (2000). hp–Finite element methods for hyperbolic problems.
  66. (2003). hp–Version discontinuous Galerkin methods with interior penalty for partial differential equations with nonnegative characteristic form.
  67. (2006). hp–version interior penalty discontinuous Galerkin finite element methods on anisotropic meshes.
  68. (1963). Interpolation and approximation. Blaisdell Publishing Co. Ginn and Co.
  69. (1995). Introduction to adaptive methods for differential equations.
  70. (1998). Local mesh design for the numerical solution of hyperbolic problems.
  71. (2006). Mathematical principles of anisotropic mesh adaptation.
  72. (1994). Measurements on a two-dimensional aerofoil with high-lift devices.
  73. (2008). Multitarget error estimation and adaptivity in aerodynamic flow simulations. doi
  74. (2001). New anisotropic a priori error estimates.
  75. (2004). New Two-Equation Turbulence Model for Aerodynamics Applications.
  76. (2008). Numerical analysis of higher order discontinuous Galerkin finite element methods.
  77. (2002). Numerical evaluation of two discontinuous Galerkin methods for the compressible Navier-Stokes equations. doi
  78. (1997). Numerical linear algebra.
  79. (2001). On residual-based a posteriori error estimation in hp-FEM.
  80. (2004). On the discontinuous Galerkin method for the numerical solution of the Navier-Stokes equations.
  81. (2005). On the error of linear interpolation and the orientation, aspect ratio, and internal angles of a triangle.
  82. (1998). p- and hp-FEM – Theory and Application to Solid and Fluid Mechanics.
  83. (2005). p-multigrid solution of high-order discontinuous Galerkin discretizations of the compressible Navier-Stokes equations.
  84. (1996). Parallel adaptive hp–refinement techniques for conservation laws.
  85. (1994). Parallel, adaptive finite element methods for conservation laws.
  86. (1990). Piecewise solenoidal vector fields and the Stokes problem. doi
  87. (2000). Review of a priori error estimation for discontinuous Galerkin methods.
  88. (1973). Second Order Equations with Nonnegative Characteristic Form.
  89. (2003). Sobolev regularity estimation for hp–adaptive finite element methods.
  90. (1975). Sobolev spaces. doi
  91. (2002). Space-time discontinuous Galerkin finite element method with dynamic grid motion for inviscid compressible flows, I. General formulation. doi
  92. (2006). Space–time discontinuous Galerkin method for the compressible Navier-Stokes equations.
  93. (2001). Stabilized hp-finite element approximation of partial differential equations with non-negative characteristic form.
  94. (2000). Stabilized hp–finite element methods for first–order hyperbolic problems.
  95. (1997). Summary of code validation results for a multiple element airfoil test case. 28th AIAA fluid dynamics conference,
  96. (2006). Symmetric interior penalty DG methods for the compressible Navier–Stokes equations I: Method formulation.
  97. (2006). Symmetric interior penalty DG methods for the compressible Navier–Stokes equations II: Goal–oriented a posteriori error estimation.
  98. (1990). The Analysis of Linear Partial Differential Operators I: Distributional Theory and Fourier Analysis.
  99. (2006). The DLR TAU-code: Recent applications in research and industry. In
  100. (1978). The finite element method for elliptic problems.
  101. (1986). The h, p and h–p versions of the finite element method in 1 dimension. Part III. The adaptive h–p version.
  102. (1987). The hp–version of the finite element method with quasiuniform meshes.
  103. (1989). Toward a universal h–p adaptive finite element strategy, Part 3. Design of h–p meshes.
  104. (2005). Toward anisotropic mesh adaptation based upon sensitivity of a posteriori estimates.
  105. (2002). Unified analysis of discontinuous Galerkin methods for elliptic problems. doi
  106. (1996). Weighted a posteriori error control in FE methods.

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