Output-Based Error Estimation and Mesh Adaptation in Computational Fluid Dynamics: Overview and Recent Results

This article reviews recent work in output-based error estimation and mesh adaptation for Computational Fluid Dynamics (CFD) applications in aerospace engineering. Output adjoint solutions, which play a central role in the error estimation, are introduced both in fully-discrete and in variational formulations. The adjoint-weighted residual method for error estimation is then presented in both formulations. Assumptions and approximations made in the calculations are discussed. Presentation of the discrete and variational formulations enables a side-by-side comparison of recent work in output error estimation using the finite volume method and the finite element method. Various mesh adaptation techniques for reducing output error are then discussed. Recent adaptive results using different implementations are presented for a variety of applications. These results show the power of output-based adaptive methods for improving the robustness of CFD computations. However, challenges and areas of additional future research, such as computable error bounds and robust mesh adaptation mechanics, exist, and these are discussed following the results.

[1]  Josep Sarrate,et al.  A POSTERIORI FINITE ELEMENT ERROR BOUNDS FOR NON-LINEAR OUTPUTS OF THE HELMHOLTZ EQUATION , 1999 .

[2]  Edward N. Tinoco,et al.  Summary of Data from the Second AIAA CFD Drag Prediction Workshop (Invited) , 2004 .

[3]  Ivo Babuška,et al.  The post‐processing approach in the finite element method—Part 2: The calculation of stress intensity factors , 1984 .

[4]  Dimitri J. Mavriplis,et al.  Adaptive mesh generation for viscous flows using delaunay triangulation , 1990 .

[5]  M. Wheeler,et al.  Anisotropic and dynamic mesh adaptation for discontinuous Galerkin methods applied to reactive transport , 2006 .

[6]  Mathias Wintzer,et al.  Adjoint-Based Adaptive Mesh Refinement for Complex Geometries , 2008 .

[7]  Paul-Louis George,et al.  Mailleur bidimensionnel de Delaunay gouverné par une carte de métriques. Partie II: Applications , 1995 .

[8]  Jonathan Richard Shewchuk,et al.  Delaunay refinement algorithms for triangular mesh generation , 2002, Comput. Geom..

[9]  D. Venditti,et al.  Adjoint error estimation and grid adaptation for functional outputs: application to quasi-one-dimensional flow , 2000 .

[10]  J. W. Purvis,et al.  Prediction of critical Mach number for store configurations , 1979 .

[11]  Ralf Hartmann,et al.  Symmetric Interior Penalty DG Methods for the Compressible Navier-Stokes Equations II: Goal--Oriented A Posteriori Error Estimation , 2005 .

[12]  Thomas Sonar,et al.  On a second order residual estimator for numerical schemes for nonlinear hyperbolic conservation laws , 2001 .

[13]  Hantaek Bae.  Navier-Stokes equations , 1992 .

[14]  Rolf Rannacher,et al.  Adaptive Galerkin finite element methods for partial differential equations , 2001 .

[15]  Steven M. Klausmeyer,et al.  Data summary from the first AIAA Computational Fluid Dynamics Drag Prediction Workshop , 2003 .

[16]  J. Eric,et al.  Aerodynamic Design Optimization on Unstructured Meshes Using the Navier-Stokes Equations , 1998 .

[17]  Antonio Huerta,et al.  The computation of bounds for linear-functional outputs of weak solutions to the two-dimensional elasticity equations , 2006 .

[18]  Dimitri J. Mavriplis,et al.  UNSTRUCTURED MESH GENERATION AND ADAPTIVITY , 1995 .

[19]  Endre Süli,et al.  The Adaptive Computation of Far-Field Patterns by A Posteriori Error Estimation of Linear Functionals , 1998 .

[20]  Juan J. Alonso,et al.  Mesh Adaptation Criteria for Unsteady Periodic Flows Using a Discrete Adjoint Time-Spectral Formulation , 2006 .

[21]  D. Scott McRae,et al.  r-Refinement grid adaptation algorithms and issues , 2000 .

[22]  Ivo Babuška,et al.  The post-processing approach in the finite element method—part 1: Calculation of displacements, stresses and other higher derivatives of the displacements , 1984 .

[23]  Rolf Jeltsch,et al.  A Higher-Order Boundary Treatment for Cartesian-Grid Methods , 1998 .

[24]  Olivier Pironneau,et al.  Mesh adaption and automatic differentiation in a CAD-free framework for optimal shape design , 1999 .

[25]  Mark Ainsworth,et al.  An adaptive refinement strategy for hp -finite element computations , 1998 .

[26]  M. Giles,et al.  Adjoint equations in CFD: duality, boundary conditions and solution behaviour , 1997 .

[27]  A. Peirce.  Computer Methods in Applied Mechanics and Engineering , 2010 .

[28]  Dominique Pelletier,et al.  Error control and mesh adaptation for the Euler equations , 2001 .

[29]  Bruno Després,et al.  Lax theorem and finite volume schemes , 2003, Math. Comput..

[30]  David L. Darmofal,et al.  Output-based Adaptive Meshing Using Triangular Cut Cells , 2006 .

[31]  E. Süli,et al.  A note on the design of hp-adaptive finite element methods for elliptic partial differential equations , 2005 .

[32]  Haihang You,et al.  Adaptive Discontinuous Galerkin Finite Element Methods , 2009 .

[33]  P. Jimack,et al.  Toward anisotropic mesh adaption based upon sensitivity of a posteriori estimates , 2022 .

[34]  W. K. Anderson,et al.  Aerodynamic design optimization on unstructured grids with a continuous adjoint formulation , 1997 .

[35]  D. Venditti,et al.  Grid adaptation for functional outputs: application to two-dimensional inviscid flows , 2002 .

[36]  Garrett. Barter,et al.  Shock Capturing with PDE-Based Artificial Viscosity for an Adaptive, Higher-Order Discontinuous Galerkin Finite Element Method , 2008 .

[37]  Michael Andrew Park,et al.  Adjoint-Based, Three-Dimensional Error Prediction and Grid Adaptation , 2002 .

[38]  Rainald Löhner,et al.  COMPARISON OF BODY-FITTED, EMBEDDED AND IMMERSED SOLUTIONS OF LOW REYNOLDS-NUMBER 3-D INCOMPRESSIBLE FLOWS , 2008 .

[39]  Mary F. Wheeler,et al.  Mesh Adaptation Strategies for Discontinuous Galerkin Methods Applied to Reactive Transport Problems , 2004 .

[40]  J. Alonso,et al.  ADjoint: An Approach for the Rapid Development of Discrete Adjoint Solvers , 2006 .

[41]  Thomas Richter,et al.  A posteriori error estimation and anisotropy detection with the dual‐weighted residual method , 2010 .

[42]  Josef Stoer,et al.  Numerische Mathematik 1 , 1989 .

[43]  Michael B. Giles,et al.  Analytic adjoint solutions for the quasi-one-dimensional Euler equations , 2001, Journal of Fluid Mechanics.

[44]  O. C. Zienkiewicz,et al.  Adaptivity and mesh generation , 1991 .

[45]  B. Wedan,et al.  A method for solving the transonic full-potential equation for general configurations , 1983 .

[46]  J. Peraire,et al.  Computing upper and lower bounds for the J-integral in two-dimensional linear elasticity , 2006 .

[47]  Michael Andrew Park,et al.  Anisotropic output-based adaptation with tetrahedral cut cells for compressible flows , 2008 .

[48]  Michael B. Giles,et al.  Solution Adaptive Mesh Refinement Using Adjoint Error Analysis , 2001 .

[49]  R. Verfiirth.  A posteriori error estimation and adaptive mesh-refinement techniques , 2001 .

[50]  M. Giles,et al.  Adjoint methods for PDEs: a posteriori error analysis and postprocessing by duality , 2002, Acta Numerica.

[51]  James Lu,et al.  An a posteriori Error Control Framework for Adaptive Precision Optimization using Discontinuous Galerkin Finite Element Method , 2005 .

[52]  C. J. Roy.  Strategies for Driving Mesh Adaptation in CFD (Invited) , 2009 .

[53]  P. D. Frank,et al.  A comparison of two closely-related approaches to aerodynamic design optimization , 1991 .

[54]  Michael B. Giles,et al.  Adjoint and defect error bounding and correction for functional estimates , 2003 .

[55]  H. Elman,et al.  DESIGN UNDER UNCERTAINTY EMPLOYING STOCHASTIC EXPANSION METHODS , 2008 .

[56]  Dimitri J. Mavriplis,et al.  Discrete Adjoint Based Time-Step Adaptation and Error Reduction in Unsteady Flow Problems , 2007 .

[57]  Richard P. Dwight,et al.  Heuristic a posteriori estimation of error due to dissipation in finite volume schemes and application to mesh adaptation , 2008, J. Comput. Phys..

[58]  Joseph H. Morrison,et al.  Statistical Analysis of CFD Solutions from the Third AIAA Drag Prediction Workshop (Invited) , 2007 .

[59]  J. Z. Zhu,et al.  The superconvergent patch recovery and a posteriori error estimates. Part 2: Error estimates and adaptivity , 1992 .

[60]  Per-Olof Persson,et al.  Curved mesh generation and mesh refinement using Lagrangian solid mechanics , 2008 .

[61]  Endre Süli,et al.  The importance of adjoint consistency in the approximation of linear functionals using the discontinuous Galerkin finite element method , 2004 .

[62]  Simona Perotto,et al.  An anisotropic a-posteriori error estimate for a convection-diffusion problem , 2001 .

[63]  Rolf Rannacher,et al.  A Feed-Back Approach to Error Control in Finite Element Methods: Basic Analysis and Examples , 1996 .

[64]  D. Darmofal,et al.  An implicit, exact dual adjoint solution method for turbulent flows on unstructured grids , 2004 .

[65]  Barna A. Szabó,et al.  Estimation and Control Error Based on P-Convergence, , 1984 .

[66]  Anthony T. Patera,et al.  Asymptotic a Posteriori Finite Element Bounds for the Outputs of Noncoercive Problems: the Helmholtz , 1999 .

[67]  M. Giles,et al.  Algorithm Developments for Discrete Adjoint Methods , 2003 .

[68]  Jaime Peraire,et al.  Implicit A-Posteriori Bounds for Discontinuous Galerkin Methods , 2003 .

[69]  Frédéric Hecht,et al.  Anisotropic unstructured mesh adaption for flow simulations , 1997 .

[70]  Mark S. Gockenbach.  Adaptive Mesh Generation , 2006 .

[71]  John C. Vassberg,et al.  Grid Generation Requirements for Accurate Drag Predictions Based on OVERFLOW Calculations , 2003 .

[72]  Gustavo C. Buscaglia,et al.  Anisotropic mesh optimization and its application in adaptivity , 1997 .

[73]  D. Venditti,et al.  Anisotropic grid adaptation for functional outputs: application to two-dimensional viscous flows , 2003 .

[74]  Mark S. Shephard,et al.  Accounting for curved domains in mesh adaptation , 2003 .

[75]  Mats G. Larson,et al.  A posteriori eror estimation for higher order Godunov finite volume methods on unstructured meshes , 2002 .

[76]  Rüdiger Verfürth,et al.  A posteriori error estimation and adaptive mesh-refinement techniques , 1994 .

[77]  Mathematisches Forschungsinstitut Oberwolfach,et al.  Hyperbolic Conservation Laws , 2004 .

[78]  J. Oden,et al.  Goal-oriented error estimation and adaptivity for the finite element method , 2001 .

[79]  Charles L. Merkle,et al.  Anisotropic Grid Adaptation on Unstructured Meshes , 2001 .

[80]  Endre Süli,et al.  hp-Adaptive Discontinuous Galerkin Finite Element Methods for First-Order Hyperbolic Problems , 2001, SIAM J. Sci. Comput..

[81]  T. Barth,et al.  Error estimation and adaptive discretization methods in computational fluid dynamics , 2003 .

[82]  M. Fortin,et al.  Anisotropic mesh adaptation: towards user‐independent, mesh‐independent and solver‐independent CFD. Part I: general principles , 2000 .

[83]  K. Duraisamy,et al.  Error Estimation for High Speed Flows Using Continuous and Discrete Adjoints , 2010 .

[84]  Ralf Hartmann,et al.  Error Estimation and Adaptive Mesh Refinement for Aerodynamic Flows , 2010 .

[85]  Thomas Sonar.  Strong and Weak Norm Refinement Indicators Based on the Finite Element Residual for Compressible Flow Computation : I. The Steady Case , 1993, IMPACT Comput. Sci. Eng..

[86]  Paul-Louis George,et al.  Mailleur bidimensionnel de Delaunay gouverné par une carte de métriques. Partie I: Algorithmes , 1995 .

[87]  Alberto Bressan.  Hyperbolic Conservation Laws , 2009, Encyclopedia of Complexity and Systems Science.

[88]  Elizabeth M. Lee-Rausch,et al.  Application of Parallel Adjoint-Based Error Estimation and Anisotropic Grid Adaptation for Three-Dimensional Aerospace Configurations , 2005 .

[89]  Rolf Rannacher,et al.  Duality-based adaptivity in the hp-finite element method , 2003, J. Num. Math..

[90]  Joseph H. Morrison.  Statistical Analysis of CFD Solutions from the Fourth AIAA Drag Prediction Workshop , 2010 .

[91]  Dimitri J. Mavriplis,et al.  Results from the 3rd Drag Prediction Workshop Using the NSU3D Unstructured Mesh Solver , 2007 .

[92]  D. D. Zeeuw,et al.  An adaptively refined Cartesian mesh solver for the Euler equations , 1993 .

[93]  Ralf Hartmann,et al.  Multitarget Error Estimation and Adaptivity in Aerodynamic Flow Simulations , 2008, SIAM J. Sci. Comput..

[94]  Endre Süli,et al.  Sobolev Regularity Estimation for hp-Adaptive Finite Element Methods , 2003 .

[95]  Kunibert G. Siebert,et al.  An a posteriori error estimator for anisotropic refinement , 1996 .

[96]  Phillip Colella,et al.  A Cartesian Grid Embedded Boundary Method for Poisson's Equation on Irregular Domains , 1998 .

[97]  Michael Andrew Park,et al.  Adjoint-Based, Three-Dimensional Error Prediction and Grid Adaptation , 2004 .

[98]  Michael A. Park,et al.  Three-Dimensional Turbulent RANS Adjoint-Based Error Correction , 2003 .

[99]  Douglas N. Arnold,et al.  Unified Analysis of Discontinuous Galerkin Methods for Elliptic Problems , 2001, SIAM J. Numer. Anal..

[100]  Jens-Dominik Müller,et al.  On the performance of discrete adjoint CFD codes using automatic differentiation , 2005 .

[101]  David L. Darmofal,et al.  An adaptive simplex cut-cell method for discontinuous Galerkin discretizations of the Navier-Stokes equations , 2007 .

[102]  Thomas Sonar,et al.  Dynamic adaptivity and residual control in unsteady compressible flow computation , 1994 .

[103]  Jaime Peraire,et al.  Computing Bounds for Linear Functionals of Exact Weak Solutions to the Advection-Diffusion-Reaction Equation , 2005, SIAM J. Sci. Comput..

[104]  J. Oden,et al.  A Posteriori Error Estimation in Finite Element Analysis , 2000 .

[105]  Edward N. Tinoco,et al.  Data Summary from Second AIAA Computational Fluid Dynamics Drag Prediction Workshop , 2003 .

[106]  T. Pulliam,et al.  Adjoint Formulation for an Embedded-Boundary Cartesian Method , 2005 .

[107]  David L. Darmofal,et al.  Parallel Anisotropic Tetrahedral Adaptation , 2008 .

[108]  J. Peraire,et al.  A posteriori finite element bounds for linear-functional outputs of elliptic partial differential equations , 1997 .

[109]  I. Babuška,et al.  Theh, p andh-p versions of the finite element method in 1 dimension , 1986 .

[110]  J. Oden,et al.  Toward a universal h - p adaptive finite element strategy: Part 2 , 1989 .

[111]  Ralf Hartmann,et al.  Adjoint Consistency Analysis of Discontinuous Galerkin Discretizations , 2007, SIAM J. Numer. Anal..

[112]  K. Powell,et al.  Solution-Adaptive Cartesian Cell Approach for Viscous and Inviscid Flows , 1996 .

[113]  J. Oden,et al.  hp-Version discontinuous Galerkin methods for hyperbolic conservation laws , 1996 .

[114]  Carl Ollivier-Gooch,et al.  Tetrahedral mesh improvement using swapping and smoothing , 1997 .

[115]  Anthony T. Patera,et al.  A posteriori finite-element output bounds for the incompressible Navier-Stokes equations: application to a natural convection problem , 2001 .

[116]  I. Babuska,et al.  A‐posteriori error estimates for the finite element method , 1978 .

[117]  Michael B. Giles,et al.  Adjoint Recovery of Superconvergent Functionals from PDE Approximations , 2000, SIAM Rev..

[118]  O. C. Zienkiewicz,et al.  Adaptive remeshing for compressible flow computations , 1987 .

[119]  Antony Jameson,et al.  OPTIMAL CONTROL OF UNSTEADY FLOWS USING A TIME ACCURATE METHOD , 2002 .

[120]  Dimitri J. Mavriplis,et al.  Unsteady Discrete Adjoint Formulation for Two-Dimensional Flow Problems with Deforming Meshes , 2008 .

[121]  I. Babuska,et al.  The h , p and h-p versions of the finite element methods in 1 dimension . Part III. The adaptive h-p version. , 1986 .

[122]  David W. Zingg,et al.  A General Framework for the Optimal Control of Unsteady Flows with Applications , 2007 .

[123]  Michael B. Giles,et al.  Improved- lift and drag estimates using adjoint Euler equations , 1999 .

[124]  I. Babuska,et al.  Theh,p andh-p versions of the finite element method in 1 dimension , 1986 .

[125]  Endre Süli,et al.  Adaptive error control for finite element approximations of the lift and drag coefficients in viscous flow , 1997 .

[126]  Timothy J. Baker,et al.  Mesh adaptation strategies for problems in fluid dynamics , 1997 .

[127]  M. D. Salas,et al.  Euler calculations for multielement airfoils using Cartesian grids , 1986 .

[128]  M. Aftosmis.  Solution adaptive cartesian grid methods for aerodynamic flows with complex geometries , 1997 .

[129]  David Anthony Venditti,et al.  Grid adaptation for functional outputs of compressible flow simulations , 2000 .

[130]  Steve L. Karman,et al.  SPLITFLOW - A 3D unstructured Cartesian/prismatic grid CFD code for complex geometries , 1995 .

[131]  Peter K. Moore,et al.  Applications of Lobatto polynomials to an adaptive finite element method: A posteriori error estimates for HP-Adaptivity and Grid-to-Grid Interpolation , 2003, Numerische Mathematik.

[132]  Juan R. Cebral,et al.  Comparison of Body-Fitted, Embedded and Immersed Solutions of Low Reynolds-Number Incompressible Flows , 2007 .

[133]  Carl Ollivier-Gooch,et al.  A generalized framework for high order anisotropic mesh adaptation , 2009 .

[134]  J. C. Newman,et al.  Adjoint-based error estimation and grid adaptation for functional outputs: Application to two-dimensional, inviscid, incompressible flows , 2009 .

[135]  Leszek Demkowicz,et al.  Toward a universal adaptive finite element strategy part 3. design of meshes , 1989 .

[136]  M. Berger,et al.  Automatic adaptive grid refinement for the Euler equations , 1985 .

[137]  R. Rannacher,et al.  A posteriori error analysis for stabilised finite element approximations of transport problems , 2000 .

[138]  Antony Jameson,et al.  Aerodynamic design via control theory , 1988, J. Sci. Comput..

[139]  David L. Darmofal,et al.  Impact of Turbulence Model Irregularity on High-Order Discretizations , 2009 .

[140]  R. Hartmann,et al.  Adaptive discontinuous Galerkin finite element methods for the compressible Euler equations , 2002 .

[141]  Rolf Rannacher,et al.  An optimal control approach to a posteriori error estimation in finite element methods , 2001, Acta Numerica.

[142]  Dimitri J. Mavriplis,et al.  An Unsteady Discrete Adjoint Formulation for Two-Dimensional Flow Problems with Deforming Meshes , 2007 .

[143]  M. Fortin,et al.  Anisotropic mesh adaptation: towards user‐independent, mesh‐independent and solver‐independent CFD. Part II. Structured grids , 2002 .

[144]  Class Johnson,et al.  On computatbility and error control in CFD , 1995 .

[145]  Paul Houston,et al.  Adaptivity and A Posteriori Error Estimation For DG Methods on Anisotropic Meshes , 2006 .

[146]  Ivo Babuška,et al.  Accuracy estimates and adaptive refinements in finite element computations , 1986 .

[147]  Michael J. Aftosmis,et al.  Multilevel Error Estimation and Adaptive h-Refinement for Cartesian Meshes with Embedded Boundaries , 2002 .

[148]  Leszek Demkowicz,et al.  Toward a universal h-p adaptive finite element strategy , 1989 .

[149]  A. Dervieux,et al.  Exact-gradient shape optimization of a 2-D Euler flow , 1992 .

[150]  David L. Darmofal,et al.  Shock Capturing with Higher-Order, PDE-Based Artificial Viscosity , 2007 .

[151]  Endre Süli,et al.  hp-Version discontinuous Galerkin methods with interior penalty for partial differential equations with nonnegative characteristic form. , 2002 .

[152]  Ralf Hartmann,et al.  Goal-Oriented A Posteriori Error Estimation for Multiple Target Functionals , 2003 .

[153]  A. U.S.,et al.  Curved Mesh Generation and Mesh Refinement using Lagrangian Solid Mechanics , 2009 .

[154]  Peter A. Gnoffo,et al.  Computational Aerothermodynamic Simulation Issues on Unstructured Grids , 2004 .

[155]  Timothy G. Trucano,et al.  Verification and Validation in Computational Fluid Dynamics , 2002 .

[156]  Claes Johnson,et al.  Numerics and hydrodynamic stability: toward error control in computational fluid dynamics , 1995 .

[157]  M. Larson,et al.  ADAPTIVE ERROR CONTROL FOR FINITE ELEMENTAPPROXIMATIONS OF THE LIFT AND DRAG COEFFICIENTS INVISCOUS , 1997 .

[158]  Marius Paraschivoiu,et al.  A posteriori finite element bounds for linear-functional outputs of coercive partial differential equations and of the Stokes problem , 1997 .

[159]  Eric Schall,et al.  Mesh adaptation as a tool for certified computational aerodynamics , 2004 .

[160]  J. Alonso,et al.  A Coupled-Adjoint Sensitivity Analysis Method for High-Fidelity Aero-Structural Design , 2005 .

[161]  Michael S. Eldred,et al.  Design Under Uncertainty Employing Stochastic Expansion Methods , 2008 .

[162]  J. Peraire,et al.  Practical Three-Dimensional Aerodynamic Design and Optimization Using Unstructured Meshes , 1997 .

[163]  Marshall Bern,et al.  Adaptive Mesh Generation , 2003 .

[164]  W. K. Anderson,et al.  Al A A-2001-0596 Recent Improvements in Aerodynamic Design Optimization On Unstructured Meshes , 2001 .

[165]  R. Verfürth,et al.  Edge Residuals Dominate A Posteriori Error Estimates for Low Order Finite Element Methods , 1999 .

[166]  A. Jameson,et al.  Optimum Shape Design for Unsteady Three-Dimensional Viscous Flows Using a Nonlinear Frequency-Domain Method , 2006 .

[167]  M. Giles,et al.  Adjoint Error Correction for Integral Outputs , 2003 .

[168]  Simona Perotto,et al.  New anisotropic a priori error estimates , 2001, Numerische Mathematik.

[169]  Leszek Demkowicz,et al.  Goal-oriented hp-adaptivity for elliptic problems , 2004 .

[170]  Ralf Hartmann,et al.  Adaptive discontinuous Galerkin methods with shock‐capturing for the compressible Navier–Stokes equations , 2006 .

[171]  Krzysztof J. Fidkowski,et al.  Output-Driven Anisotropic Mesh Adaptation for Viscous Flows using Discrete Choice Optimization , 2010 .

[172]  Richard H. Burkhart,et al.  A new approach to the solution of boundary value problems involving complex configurations , 1986 .

[173]  Dominique Pelletier,et al.  Mesh adaptation using different error indicators for the Euler equations , 2001 .

[174]  Roger W. Logan,et al.  Comparing 10 Methods for Solution Verification, and Linking to Model Validation , 2005, J. Aerosp. Comput. Inf. Commun..

[175]  Leszek F. Demkowicz,et al.  A Fully Automatic hp-Adaptivity , 2002, J. Sci. Comput..

[176]  G. Strang,et al.  An Analysis of the Finite Element Method , 1974 .

[177]  Tobias Leicht.  Anisotropic Mesh Refinement for Discontinuous Galerkin Methods in Aerodynamic Flow Simulations , 2006 .

[178]  Serge Prudhomme,et al.  New approaches to error estimation and adaptivity for the Stokes and Oseen equations , 1999 .

[179]  Antonio Huerta,et al.  Computing Bounds for Linear Functionals of Exact Weak Solutions to Poisson's Equation , 2004, SIAM J. Numer. Anal..

[180]  Todd A. Oliver.  A High-Order, Adaptive, Discontinuous Galerkin Finite Element Method for the Reynolds-Averaged Navier-Stokes Equations , 2008 .

[181]  J. Newman,et al.  Comparison of adjoint‐based and feature‐based grid adaptation for functional outputs , 2007 .

[182]  David L. Darmofal,et al.  A Quasi-Minimal Residual Method for Simultaneous Primal-Dual Solutions and Superconvergent Functional Estimates , 2002, SIAM J. Sci. Comput..

[183]  Eric J. Nielsen,et al.  Validation of 3D Adjoint Based Error Estimation and Mesh Adaptation for Sonic Boom Prediction , 2006 .

[184]  R. Temam.  Navier-Stokes Equations , 1977 .

[185]  J. Z. Zhu,et al.  The superconvergent patch recovery and a posteriori error estimates. Part 1: The recovery technique , 1992 .

[186]  Kenneth G. Powell,et al.  An adaptively-refined Cartesian mesh solver for the Euler equations , 1991 .

[187]  William L. Kleb,et al.  On Multi-Dimensional Unstructured Mesh Adaption , 1999 .

[188]  David L. Darmofal,et al.  A triangular cut-cell adaptive method for high-order discretizations of the compressible Navier-Stokes equations , 2007, J. Comput. Phys..

[189]  Antony Jameson,et al.  Optimum Shape Design for Unsteady Flows with Time-Accurate Continuous and Discrete Adjoint Methods , 2007 .