Error Control, Adaptive Discretizations, and Applications, Volume 58, Part One highlights new advances in the field, with this new volume presenting interesting chapters written by an international board of authors. Chapters in this release cover hp adaptive Discontinuous Galerkin strategies driven by a posteriori error estimation with application to aeronautical flow problems,
An anisotropic mesh adaptation method based on gradient recovery and optimal shape elements, and Model reduction techniques for parametrized nonlinear partial differential equations.
1. 1 Introduction - p. 38 2. 2 Governing equations and turbulence modeling - p. 41 3. 3 Numerical methodology - p. 47 4. 4 A posteriori error estimation - p. 53 5. 5 Adaptation strategy - p. 58 6. 6 Validation - p. 61 7. 7 Applications - p. 70 8. 8 Outcomes and perspectives - p. 93 9. References - p. 100 10. Chapter Three: An anisotropic mesh adaptation method based on gradient recovery and optimal shape elements - p. 101 11. Abstract - p. 101 12. 1 Introduction - p. 102 13. 2 Notation and classical error estimates - p. 104 14. 3 Definition of an optimal mesh - p. 105 15. 4 Construction of an enriched solution and error estimation - p. 110 16. 5 Local operations on the mesh - p. 119 17. 6 Academic examples - p. 130 18. 7 Applications - p. 134 19. 8 Conclusions - p. 142 20. Acknowledgments - p. 144 21. References - p. 148 22. Chapter Four: Model reduction techniques for parametrized nonlinear partial differential equations - p. 149 23. Abstract - p. 149 24. 1 Introduction - p. 150 25. 2 Hyper-reduction methods - p. 152 26. 3 First-order empirical interpolation method - p. 156 27. 4 Projection-based model reduction methods - p. 169 28. 5 Nonintrusive model reduction techniques - p. 180 29. 6 Concluding remarks - p. 192 30. Acknowledgement - p. 196 31. References - p. 204 32. Chapter Five: Adaptive finite elements for obstacle problems - p. 2061 33. Chapter Six: A posteriori error identities and estimates of modelling errors - p. 245 34. Abstract - p. 245 35. 1 Introduction - p. 246 36. 2 Mathematical background - p. 251 37. 3 Error identities - p. 256 38. 4 Simplification of coefficients - p. 262 39. 5 Errors stipulated by geometric simplifications - p. 269 40. 6 Dimension reduction errors - p. 275 41. References - p. 293 42. Chapter Seven: Exact a posteriori error control for variational problems via convex duality and explicit flux reconstruction - p. 295 43. Abstract - p. 295 44. 1 Introduction - p. 296 45. 2 Preliminaries - p. 310 46. 3 Exact a posteriori error estimation for convex minimization problems - p. 317 47. 4 Model problems - p. 328 48. 5 Equivalence to residual type error estimators - p. 341 49. 6 Quasi-optimality of node-averaging operator - p. 346 50. 7 Numerical experiments - p. 348 51. 8 Conclusion - p. 369 52. References - p. 376 53. Chapter Eight: Algebraic error in numerical PDEs and its estimation - p. 377 54. Abstract - p. 377 55. 1 Introduction - p. 378 56. 2 Model problem and notation - p. 380 57. 3 Errors of different origin (and different behavior) - p. 381 58. 4 Residual-based error estimators and adaptive mesh refinement - p. 385 59. 5 Error estimates based on flux reconstructions—all-in-one - p. 397 60. 6 Conclusions - p. 423 61. Acknowledgment - p. 424 62. References - p. 427
Error Control, Adaptive Discretizations, and Applications, Volume 58, Part One highlights new advances in the field, with this new volume presenting interesting chapters written by an international board of authors. Chapters in this release cover hp adaptive Discontinuous Galerkin strategies driven by a posteriori error estimation with application to aeronautical flow problems,
An anisotropic mesh adaptation method based on gradient recovery and optimal shape elements, and Model reduction techniques for parametrized nonlinear partial differential equations.
1. 1 Introduction - p. 38 2. 2 Governing equations and turbulence modeling - p. 41 3. 3 Numerical methodology - p. 47 4. 4 A posteriori error estimation - p. 53 5. 5 Adaptation strategy - p. 58 6. 6 Validation - p. 61 7. 7 Applications - p. 70 8. 8 Outcomes and perspectives - p. 93 9. References - p. 100 10. Chapter Three: An anisotropic mesh adaptation method based on gradient recovery and optimal shape elements - p. 101 11. Abstract - p. 101 12. 1 Introduction - p. 102 13. 2 Notation and classical error estimates - p. 104 14. 3 Definition of an optimal mesh - p. 105 15. 4 Construction of an enriched solution and error estimation - p. 110 16. 5 Local operations on the mesh - p. 119 17. 6 Academic examples - p. 130 18. 7 Applications - p. 134 19. 8 Conclusions - p. 142 20. Acknowledgments - p. 144 21. References - p. 148 22. Chapter Four: Model reduction techniques for parametrized nonlinear partial differential equations - p. 149 23. Abstract - p. 149 24. 1 Introduction - p. 150 25. 2 Hyper-reduction methods - p. 152 26. 3 First-order empirical interpolation method - p. 156 27. 4 Projection-based model reduction methods - p. 169 28. 5 Nonintrusive model reduction techniques - p. 180 29. 6 Concluding remarks - p. 192 30. Acknowledgement - p. 196 31. References - p. 204 32. Chapter Five: Adaptive finite elements for obstacle problems - p. 2061 33. Chapter Six: A posteriori error identities and estimates of modelling errors - p. 245 34. Abstract - p. 245 35. 1 Introduction - p. 246 36. 2 Mathematical background - p. 251 37. 3 Error identities - p. 256 38. 4 Simplification of coefficients - p. 262 39. 5 Errors stipulated by geometric simplifications - p. 269 40. 6 Dimension reduction errors - p. 275 41. References - p. 293 42. Chapter Seven: Exact a posteriori error control for variational problems via convex duality and explicit flux reconstruction - p. 295 43. Abstract - p. 295 44. 1 Introduction - p. 296 45. 2 Preliminaries - p. 310 46. 3 Exact a posteriori error estimation for convex minimization problems - p. 317 47. 4 Model problems - p. 328 48. 5 Equivalence to residual type error estimators - p. 341 49. 6 Quasi-optimality of node-averaging operator - p. 346 50. 7 Numerical experiments - p. 348 51. 8 Conclusion - p. 369 52. References - p. 376 53. Chapter Eight: Algebraic error in numerical PDEs and its estimation - p. 377 54. Abstract - p. 377 55. 1 Introduction - p. 378 56. 2 Model problem and notation - p. 380 57. 3 Errors of different origin (and different behavior) - p. 381 58. 4 Residual-based error estimators and adaptive mesh refinement - p. 385 59. 5 Error estimates based on flux reconstructions—all-in-one - p. 397 60. 6 Conclusions - p. 423 61. Acknowledgment - p. 424 62. References - p. 427
