Alain Dervieux is chief scientist at the Société Lemma and emeritus senior scientist at Inria, Sophia Antipolis. His main research interests are computational fluid dynamics, particularly approximations on unstructured meshes.Frederic Alauzet is a senior researcher at Inria Saclay and adjunct professor at Mississippi State University. His research focuses on anisotropic mesh adaptation, advanced solvers, mesh generation and moving mesh methods.Adrien Loseille is a research scientist at Inria Saclay, working in Luminary Cloud. His main domains of interest are unstructured mesh generation and adaptation for computational fluid dynamics.Bruno Koobus is professor at the University of Montpellier. His main research interests cover computational fluid dynamics, in particular the development of numerical methods on fixed and moving meshes, turbulence modeling and parallel algorithms.

• Acknowledgments ixIntroduction xiChapter 1 Nonlinear Corrector for CFD 11.1. Introduction 11.1.1. Linear correction 31.1.2. Nonlinear correction 41.2. Two correctors for the Poisson problem 51.2.1. Notations 51.2.2. A priori corrector for the PDE solution 61.2.3. Finer-grid DC corrector for the PDE solution 81.3. RANS equations 91.3.1. Vector form of the RANS system 91.3.2. Formal discretization 101.3.3. Notations for discretization 111.4. Nonlinear functional correction 131.4.1. Finite volume nonlinear corrector 131.4.2. Finite element corrector 151.5. Example: supersonic flow 171.6. Concluding remarks 181.7. Notes 20Chapter 2 Multi-scale Adaptation for Unsteady Flows 212.1. Introduction 212.2. Mesh adaptation efficiency 232.2.1. Regular and singular unsteady model 232.2.2. Representativity of the spatial interpolation error 242.3. Transient fixed-point mesh adaptation scheme 252.3.1. Size of subintervals in a mesh convergence 282.3.2. Mesh adaptation for unsteady Euler/Navier–Stokes equations with thickened interface 292.3.3. Convergent transient fixed-point 332.4. 2D bi-fluid example 332.5. Example: impact of a 3D water column on a obstacle 352.6. Conclusion 392.7. Appendix: remarks about the adaptation of the time step 392.8. Notes 41Chapter 3 Multi-rate Time Advancing 433.1. Introduction 433.2. Multi-rate time advancing by volume agglomeration 453.2.1. Finite volume Navier–Stokes 453.2.2. Inner and outer zones 463.2.3. MR time advancing 473.3. Elements of analysis 493.3.1. Stability 493.3.2. Accuracy 503.3.3. Efficiency 513.3.4. Toward many rates 523.3.5. Impact of our MR complexity on mesh adaption 523.3.6. Parallelism 533.4. Applications 553.4.1. Circular cylinder at very high Reynolds number 553.4.2. Mesh adaption for a contact discontinuity 583.5. Conclusion 593.6. Notes 60Chapter 4 Goal-Oriented Adaptation for Inviscid Steady Flows 654.1. Introduction 654.1.1. What to do with this estimate? 674.1.2. Adjoint-L 1 approach 684.1.3. Outline 694.2. A more accurate nonlinear error analysis 694.2.1. Assumptions and definitions 694.2.2. A priori estimation 704.3. The case of the steady Euler equations 724.3.1. Variational analysis 724.3.2. Approximation error estimation 734.4. Error model minimization 744.5. Adaptative strategy 764.5.1. Adjoint solver 774.5.2. Optimal goal-oriented discrete metric 774.5.3. Controlled mesh regeneration 794.6. Numerical outputs 794.6.1. High-fidelity pressure prediction of an aircraft 794.7. Conclusion 824.8. Notes 82Chapter 5. Goal-Oriented Adaptation for Viscous Steady Flows 855.1. Introduction 855.2. Case of an elliptic problem 865.2.1. A priori finite-element analysis (first estimate) 865.2.2. Goal-oriented adaptation according to lemma 5.1 895.2.3. Goal-oriented adaptation according to a second estimate 915.3. Error analysis for Navier–Stokes problem 925.3.1. Mesh adaptation problem statement 925.3.2. Linearized error system 935.3.3. First estimate for Navier–Stokes problem 945.3.4. Second estimate for Navier–Stokes problem 985.3.5. Optimal goal-oriented continuous mesh 1015.4. From theory to practice 1015.4.1. Computation of the optimal continuous mesh 1035.5. An example of application to a turbulent flow 1035.6. Conclusion 1075.7. Notes 109Chapter 6 Norm-Oriented Formulations 1116.1. Introduction 1116.2. A summary of previous analyses 1146.2.1. Feature-based adaptation by interpolation error optimization 1146.2.2. Implicit a priori error estimate and corrector 1156.2.3. Goal-oriented analysis 1166.3. Norm-oriented approach 1186.4. Numerical elliptic examples 1196.4.1. Numerical features 1196.4.2. 2D boundary layer 1226.4.3. Poisson problem with discontinuous coefficient 1236.5. Application to flows 1266.5.1. A comparison feature-oriented/norm 1276.5.2. Application to a viscous flow 1296.6. Conclusion 1306.7. Notes 131Chapter 7 Goal-Oriented Adaptation for Unsteady Flows 1337.1. Introduction 1337.2. Formal error analysis 1347.3. Unsteady Euler models 1357.3.1. Continuous state system and finite volume formulation 1357.3.2. Continuous adjoint system and discretization 1377.3.3. Impact of the adjoint: numerical example 1417.4. Optimal unsteady adjoint-based metric 1427.4.1. Error analysis for the unsteady Euler model 1427.4.2. Continuous error model 1447.4.3 Spatial minimization for a fixed t 1467.4.4. Temporal minimization 1467.4.5. Temporal minimization for time sub-intervals 1507.5. Theoretical mesh convergence analysis 1557.5.1. Smooth flow fields 1557.6. From theory to practice 1577.6.1. Choice of the GO metric 1587.6.2. Global fixed-point mesh adaptation algorithm 1587.6.3. Computing the GO metric 1617.7. Numerical experiments 1617.7.1. 2D Acoustic wave propagation 1617.7.2. 3D blast wave propagation 1637.8. Conclusion 1657.9. Notes 166Chapter 8 Third-Order Unsteady Adaptation 1678.1. Introduction 1678.2. Higher order interpolation and reconstruction 1688.3. CENO approximation for the 2D Euler equations 1708.3.1. Model 1708.3.2. CENO formulation 1718.3.3. Vertex-centered low dissipation CENO 2 1748.4. Error analysis 1758.5. Metric-based error estimate 1788.6. Optimal metric 1798.7. From theory to practical application 1828.8. A numerical example: acoustic wave 1838.9. Conclusion 1868.10. Notes 186References 189Index 199Summary of Volume 1 201

Alain Dervieux is chief scientist at the Société Lemma and emeritus senior scientist at Inria, Sophia Antipolis. His main research interests are computational fluid dynamics, particularly approximations on unstructured meshes.Frederic Alauzet is a senior researcher at Inria Saclay and adjunct professor at Mississippi State University. His research focuses on anisotropic mesh adaptation, advanced solvers, mesh generation and moving mesh methods.Adrien Loseille is a research scientist at Inria Saclay, working in Luminary Cloud. His main domains of interest are unstructured mesh generation and adaptation for computational fluid dynamics.Bruno Koobus is professor at the University of Montpellier. His main research interests cover computational fluid dynamics, in particular the development of numerical methods on fixed and moving meshes, turbulence modeling and parallel algorithms.

• Acknowledgments ixIntroduction xiChapter 1 Nonlinear Corrector for CFD 11.1. Introduction 11.1.1. Linear correction 31.1.2. Nonlinear correction 41.2. Two correctors for the Poisson problem 51.2.1. Notations 51.2.2. A priori corrector for the PDE solution 61.2.3. Finer-grid DC corrector for the PDE solution 81.3. RANS equations 91.3.1. Vector form of the RANS system 91.3.2. Formal discretization 101.3.3. Notations for discretization 111.4. Nonlinear functional correction 131.4.1. Finite volume nonlinear corrector 131.4.2. Finite element corrector 151.5. Example: supersonic flow 171.6. Concluding remarks 181.7. Notes 20Chapter 2 Multi-scale Adaptation for Unsteady Flows 212.1. Introduction 212.2. Mesh adaptation efficiency 232.2.1. Regular and singular unsteady model 232.2.2. Representativity of the spatial interpolation error 242.3. Transient fixed-point mesh adaptation scheme 252.3.1. Size of subintervals in a mesh convergence 282.3.2. Mesh adaptation for unsteady Euler/Navier–Stokes equations with thickened interface 292.3.3. Convergent transient fixed-point 332.4. 2D bi-fluid example 332.5. Example: impact of a 3D water column on a obstacle 352.6. Conclusion 392.7. Appendix: remarks about the adaptation of the time step 392.8. Notes 41Chapter 3 Multi-rate Time Advancing 433.1. Introduction 433.2. Multi-rate time advancing by volume agglomeration 453.2.1. Finite volume Navier–Stokes 453.2.2. Inner and outer zones 463.2.3. MR time advancing 473.3. Elements of analysis 493.3.1. Stability 493.3.2. Accuracy 503.3.3. Efficiency 513.3.4. Toward many rates 523.3.5. Impact of our MR complexity on mesh adaption 523.3.6. Parallelism 533.4. Applications 553.4.1. Circular cylinder at very high Reynolds number 553.4.2. Mesh adaption for a contact discontinuity 583.5. Conclusion 593.6. Notes 60Chapter 4 Goal-Oriented Adaptation for Inviscid Steady Flows 654.1. Introduction 654.1.1. What to do with this estimate? 674.1.2. Adjoint-L 1 approach 684.1.3. Outline 694.2. A more accurate nonlinear error analysis 694.2.1. Assumptions and definitions 694.2.2. A priori estimation 704.3. The case of the steady Euler equations 724.3.1. Variational analysis 724.3.2. Approximation error estimation 734.4. Error model minimization 744.5. Adaptative strategy 764.5.1. Adjoint solver 774.5.2. Optimal goal-oriented discrete metric 774.5.3. Controlled mesh regeneration 794.6. Numerical outputs 794.6.1. High-fidelity pressure prediction of an aircraft 794.7. Conclusion 824.8. Notes 82Chapter 5. Goal-Oriented Adaptation for Viscous Steady Flows 855.1. Introduction 855.2. Case of an elliptic problem 865.2.1. A priori finite-element analysis (first estimate) 865.2.2. Goal-oriented adaptation according to lemma 5.1 895.2.3. Goal-oriented adaptation according to a second estimate 915.3. Error analysis for Navier–Stokes problem 925.3.1. Mesh adaptation problem statement 925.3.2. Linearized error system 935.3.3. First estimate for Navier–Stokes problem 945.3.4. Second estimate for Navier–Stokes problem 985.3.5. Optimal goal-oriented continuous mesh 1015.4. From theory to practice 1015.4.1. Computation of the optimal continuous mesh 1035.5. An example of application to a turbulent flow 1035.6. Conclusion 1075.7. Notes 109Chapter 6 Norm-Oriented Formulations 1116.1. Introduction 1116.2. A summary of previous analyses 1146.2.1. Feature-based adaptation by interpolation error optimization 1146.2.2. Implicit a priori error estimate and corrector 1156.2.3. Goal-oriented analysis 1166.3. Norm-oriented approach 1186.4. Numerical elliptic examples 1196.4.1. Numerical features 1196.4.2. 2D boundary layer 1226.4.3. Poisson problem with discontinuous coefficient 1236.5. Application to flows 1266.5.1. A comparison feature-oriented/norm 1276.5.2. Application to a viscous flow 1296.6. Conclusion 1306.7. Notes 131Chapter 7 Goal-Oriented Adaptation for Unsteady Flows 1337.1. Introduction 1337.2. Formal error analysis 1347.3. Unsteady Euler models 1357.3.1. Continuous state system and finite volume formulation 1357.3.2. Continuous adjoint system and discretization 1377.3.3. Impact of the adjoint: numerical example 1417.4. Optimal unsteady adjoint-based metric 1427.4.1. Error analysis for the unsteady Euler model 1427.4.2. Continuous error model 1447.4.3 Spatial minimization for a fixed t 1467.4.4. Temporal minimization 1467.4.5. Temporal minimization for time sub-intervals 1507.5. Theoretical mesh convergence analysis 1557.5.1. Smooth flow fields 1557.6. From theory to practice 1577.6.1. Choice of the GO metric 1587.6.2. Global fixed-point mesh adaptation algorithm 1587.6.3. Computing the GO metric 1617.7. Numerical experiments 1617.7.1. 2D Acoustic wave propagation 1617.7.2. 3D blast wave propagation 1637.8. Conclusion 1657.9. Notes 166Chapter 8 Third-Order Unsteady Adaptation 1678.1. Introduction 1678.2. Higher order interpolation and reconstruction 1688.3. CENO approximation for the 2D Euler equations 1708.3.1. Model 1708.3.2. CENO formulation 1718.3.3. Vertex-centered low dissipation CENO 2 1748.4. Error analysis 1758.5. Metric-based error estimate 1788.6. Optimal metric 1798.7. From theory to practical application 1828.8. A numerical example: acoustic wave 1838.9. Conclusion 1868.10. Notes 186References 189Index 199Summary of Volume 1 201