Numerical Models for Differential Problems

Modeling, Simulation and Applications

Volumn 2, Issue , 2009, Pages 1-593

  Quarteroni, Alfio ^a,b  

^a POLITECNICO DI MILANO   (Italy)

^b EPFL   (Switzerland)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 79952253048     PISSN: 20375255     EISSN: 20375263     Source Type: Book Series    
DOI: 10.1007/978-88-470-1071-0     Document Type: Article

References (252)

1. Adams R. A. (1975) Sobolev Spaces. Academic Press, New York.
2. Almeida R. C., Feijóo R. A., Galeão A. C., Padra C., and Silva R. S. (2000) Adaptive finite element computational fluid dynamics using an anisotropic error estimator. Comput. Methods Appl. Mech. Engrg. 182: 379-400.
3. Agoshkov V. (2003) Optimal Control Methods and Adjoint Equations in Mathematical Physics Problems. Institute of Numerical Mathematics, Russian Academy of Science, Moscow.
4. Akin J. E. (1994) Finite Elements for Analysis and Design. Academic Press, London.
5. Ainsworth M. and Oden J. T. (2000) A posteriori error estimation in finite element analysis. Pure and AppliedMathematics. JohnWiley and Sons, New York.
6. Apel T. (1999) Anisotropic Finite Elements: Local Estimates and Applications. Book Series: Advances in Numerical Mathematics. Teubner, Stuttgart.
7. Achdou Y., Pironneau O., and Valentin F. (1998) Équations aux dérivées partielles et applications, chapter Shape control versus boundary control, pages 1-18. Éd. Sci. Méd. Elsevier, Paris.
8. Arpaci V. S. (1966) Conduction heat transfer. Addison-Wesley, New York.
9. Allen D. N. G. and Southwell R.V. (1955) Relaxationmethods applied to determine the motion, in two dimensions, of a viscous fluid past a fixed cylinder. Quart. J. Mech. Appl. Math. 8: 129-145.
10. Adalsteinsson D. and Sethian J. A. (1999) The fast construction of extension velocities in level set methods. J. Comput. Phys. 148(1): 2-22.
11. Almroth B. O., Stern P., and Brogan F. A. (1978) Automatic choice of global shape functions in structural analysis. AIAA Journal 16: 525-528.
12. Alekseev V., Tikhominov V., and Fomin S. (1987) Optimal Control. Consultants Bureau, New York.
13. Aubin J. P. (1967) Behavior of the error of the approximate solutions of boundary value problems for linear elliptic operators by Galerkin's and finite difference methods. Ann. Scuola Norm. Sup. Pisa 21: 599-637.
14. Aziz A., Wingate J., and Balas M. (1971) Control Theory of Systems Governed by Partial Differential Equations. Academic Press, New York.
15. www.cs.sandia.gov/CRF/aztec1.html.
16. Babuška I. (1971) Error bounds for the finite element method. Numer. Math. 16: 322-333.
17. Babuška I., Durán R., and Rodríguez R. (1992) Analysis of the efficiency of an a posteriori error estimator for linear triangular finite elements. SIAM J. Numer. Anal. 29(4): 947-964.
18. Bern M. and Eppstein D. (1992) Mesh generation and optimal triangulation. In Du D.-Z. and Hwang F. (eds) Computing in Euclidean Geometry. World Scientific, Singapore.
19. Becker R. (2001)Mesh adaptation for stationary flow control. J. Math. Fluid Mech. 3: 317-341.
20. Brezzi F. and Fortin M. (1991) Mixed and Hybrid Finite Element Methods. Springer-Verlag, New York.
21. Brezzi F. and Fortin M. (1991) Mixed and Hybrid Finite Element Methods, volume 15 of Springer Series in Computational Mathematics. Springer-Verlag, New York.
22. Brezzi F., Franca L. P., Hughes T. J. R., and Russo A. (1997) b = ∫ g. Comput. Methods Appl. Mech. Engrg. 145: 329-339.
23. Brezzi F. and Gilardi G. (1987) Functional Analysis and Functional Spaces. Mc-Graw Hill, New York.
24. Bernardi C. and Girault V. (1998) A local regularisation operator for triangular and quadrilateral finite elements. SIAM J. Numer. Anal. 35(5): 1893-1916.
25. Benzi M., Golub G. H., and Liesen J. (2005) Numerical solution of saddle-point problems. Acta Numer. 14: 1-137.
26. Bjòrstad P., Gropp P., and Smith B. (1996) Domain Decomposition, Parallel Multilevel Methods for Elliptic Partial Differential Equations. Univ. Cambridge Press, Cambridge.
27. Berselli L. C., Iliescu T., and Layton W. J. (2006) Mathematics of Large Eddy Simulation of Turbulent Flows. Springer, Berlin Heidelberg.
28. Becker R., Kapp H., and Rannacher R. (2000) Adaptive finite element methods for optimal control of partial differential equations: Basic concepts. SIAM, J. Control Opt. 39(1): 113-132.
29. Blanckaert K. (2002) Flow and turbulence in sharp open-channel bends. PhD thesis, École Polytechnique Fédérale de Lausanne.
30. Bernardi C. and Maday Y. (1992) Approximations Spectrales de Problèmes aux Limites Elliptiques. Springer-Verlag, Berlin Heidelberg.
31. Bottasso C. L., Maisano G., Micheletti S., and Perotto S. (2006) On some new recovery based a posteriori error estimators. Comput. Methods Appl. Mech. Engrg. 195(37-40): 4794-4815.
32. Brezzi F., Marini L. D., and Süli E. (2004) DiscontinuosGalerkin methods for firstorder hyperbolic problems. Math. Models Methods Appl. Sci. 14: 1893-1903.
33. Boland J. and Nicolaides R. (1983) Stability of finite elements under divergence constraints. SIAM J. Numer. Anal. 20: 722-731.
34. Barrault M., Nguyen N. C., Maday Y., and Patera A. T. (2004) An "empirical interpolation" method: Application to efficient reduced-basis discretization of partial differential equations. C. R. Acad. Sci. Paris, Série I. 339: 667-672.
35. Benzi M. and Olshanskii M. (2006) An augmented lagrangian-based approach to the oseen problem. SIAM J. on Scientific Computing 28 (6): 2095-2113.
36. Barrett A. and Reddien G. (1995) On the reduced basis method. Z. Angew. Math. Mech. 75(7): 543-549.
37. Brezzi F. (1974) On the existence, uniqueness and approximation of saddle-point problems arising from Lagrange multipliers. R.A.I.R.O. Anal. Numér. 8: 129-151.
38. Brezis H. (1986) Analisi Funzionale. Liguori, Napoli.
39. Bressan A. (2000) Hyperbolic Systems of Conservation Laws: The Onedimensional Cauchy Problem. Oxford Lecture Series in Mathematics and its Applications. The Clarendon Press Oxford University Press, New York.
40. Brenner S. C. and Scott L. R. (1994) The Mathematical Theory of Finite Element Methods. Springer-Verlag, New York.
41. Bui-Thanh T., Damodaran M., and Willcox K. (2003) Proper orthogonal decomposition extensions for parametric applications in transonic aerodynamics (AIAA Paper 2003-4213). In Proceedings of the 15th AIAA Computational Fluid Dynamics Conference.
42. Caboussat A. (2003) Analysis and numerical simulation of free surface flows. PhD thesis, École Polytechnique Fédérale de Lausanne.
43. Cazemier W. (1997) Proper Orthogonal Decomposition and Low Dimensional Models for Turbolent Flows. University of Groningen.
44. Christensen E., Bròns M., and Sòrensen J. (2000) Evaluation of proper orthogonal decomposition-based decomposition techniques applied to parameter-dependent nonturbulent flows. SIAM J. Scientific Computing 21(4): 1419-1434.
45. Çengel Y. (2007) Introduction to Thermodynamics and heat transfer. McGraw-Hill, New York.
46. Collis S. and Heinkenschloss M. (2001) Analysis of the streamline upwind/petrov galerkin method applied to the solution of optimal control problems. CAAM report TR02-01.
47. Canuto C., Hussaini M., Quarteroni A., and Zang T. A. (2006) Spectral Methods. Fundamentals in Single Domains. Springer-Verlag, Berlin Heidelberg.
48. Canuto C., Hussaini M. Y., Quarteroni A., and Zang T. A. (2007) SpectralMethods. Evolution to Complex Geometries and Application to Fluid Dynamics. Springer-Verlag, Berlin Heidelberg.
49. Ciarlet P. G. (1978) The Finite Element Method for Elliptic Problems. North-Holland, Amsterdam.
50. Cohen G., Joly P., Roberts J. E., and Tordjman N. (2001) Higher order triangular finite elements with mass lumping for the wave equation. SIAM J. Numer. Anal. 38(6): 2047-2078 (electronic).
51. Clément P. (1975) Approximation by finite element functions using local regularization. RAIRO, Anal. Numér 2 pages 77-84.
52. Cockburn B. (1998) An introduction to the discontinuous Galerkin method for convection-dominated problems. In Quarteroni A. (ed) Advanced Numerical Approximation of Nonlinesr Hyperbolic Equations, volume 1697 of LNM, pages 151-268. Springer-Verlag, Berlin Heidelberg.
53. Comincioli V. (1995) Analisi Numerica. Metodi Modelli Applicazioni. McGraw-Hill, Milano.
54. Cremonesi P. and Psaila G. (2000) Introduzione ragionata al C/C++. Progetto Leonardo, Esculapio, Bologna.
55. C. Zienkiewicz O. and Zhu J. Z. (1987) A simple error estimator and adaptive procedure for practical engineering analysis. Int. J. Numer. Meth. Engng. 24: 337-357.
56. Dedé L. (2004)Controllo Ottimale e Adattività per Equazioni alle DerivateParziali e Applicazioni. Tesi di Laurea, Politecnico di Milano.
57. Deparis S. (2008) Reduced basis error bound computation of parameter-dependent Navier-Stokes equations by the natural normapproach. SIAM Journal of Numerical Analysis 46(4): 2039-2067.
58. Detomi D., Parolini N., and Quarteroni A. (2008) Mathematics in the wind. MOX Reports (25). see the web page http://mox.polimi.it.
59. Dedè L. and Quarteroni A. (2005) Optimal control and numerical adaptivity for advection-diffusion equations. M2AN Math. Model. Numer. Anal. 39(5): 1019-1040.
60. Dervieux A. and Thomasset F. (1980) Approximation Methods for Navier-Stokes Problems, volume 771 of Lecture Notes in Mathematics, chapter A finite element method for the simulation of Rayleigh-Taylor instability, pages 145-158. Springer-Verlag, Berlin.
61. Dubiner M. (1991) Spectral methods on triangles and other domains. J. Sci. Comput. 6: 345-390.
62. Darmofal D. L. and Venditti D. A. (2002) Grid adaptation for functional outputs: application to two-dimensional inviscid flows. J. Comput. Phys. 176: 40-69.
63. DiPietro D. A. and Veneziani A. (2008) Expression templates implementation of continuous and discontinuous Galerkin methods. Computing and Visualization in Science 12.
64. Dáger R. and Zuazua E. (2006)Wave Propagation, Observation and Control in 1-d Flexible Multi-Structures. Mathématiques et Applications. Springer, Paris.
65. Ern A. and Guermond J. L. (2004) Theory and Practice of Finite Elements, volume 159 of Applied Mathematics Sciences. Springer-Verlag, New York.
66. Elman H. C., Howte V. E., Shadid J., Shuttleworth R., and Tuminaro R. (2006) Block preconditioners based on approximate commutators. SIAM J. on Scientific Computing 27 (5): 1651-1668.
67. Eriksson E. and Johnson C. (1988) An adaptive method for linear elliptic problems. Math. Comp. 50: 361-383.
68. Embree M. (1999) Convergence of Krylov subspace methods for non-normal matrices. PhD thesis, Oxford University Computing Laboratories.
69. Embree M. (2003) The tortoise and the hare restart GMRES. SIAM Rev. 45 (2): 259-266.
70. Elman H., Silvester D., and Wathen A. (2005) Finite Elements and Fast Iterative Solvers. Oxford Science Publications, Oxford.
71. Evans L. (1998) Partial differential equations. American Mathematical Society, Providence.
72. Fernández-Cara E. and Zuazua E. (2003) Control theory: History, mathematical achievements and perspectives. Bol. Soc. Esp. Mat. Apl. 26: 79-140.
73. Fernández-Cara E. and Zuazua E. (2004) On the history and perspectives of control theory. Matapli 74: 47-73.
74. Fox R. L. and Miura H. (1971) An approximate analysis technique for design calculations. AIAA Journal 9(1): 177-179.
75. Formaggia L., Micheletti S., and Perotto S. (2004) Anisotropic mesh adaptation in computational fluid dynamics: application to the advection-diffusion-reaction and the stokes problems. Appl. Numer. Math. 51(4): 511-533.
76. Foias C., Manley O., Rosa R., and Temam R. (2001) Navier-Stokes Equations and Turbulence. Cambridge Univ. Press, Cambridge.
77. Fortin M. (1977) An analysis of the convergence of mixed finite element methods. R.A.I.R.O. Anal. Numér. 11.
78. Formaggia L. and Perotto S. (2001) New anisotropic a priori error estimates. Numer. Math. 89: 641-667.
79. Ferziger J. H. and Peric M. (2002) Computational Methods for Fluid Dynamics. Springer, Berlin, III edition.
80. Fink J. P. and Rheinboldt W. C. (1983) On the error behavior of the reduced basis technique for nonlinear finite element approximations. Z. Angew. Math. Mech. 63(1): 21-28.
81. Formaggia L., Saleri F., and Veneziani A. (2005) Applicazioni ed esercizi di modellistica numerica per problemi differenziali. Springer Italia, Milano.
82. Funaro D. (1992) Polynomial Approximation of Differential Equations. Springer-Verlag, Berlin Heidelberg.
83. Funaro D. (1997) SpectralElements for Transport-DominatedEquations. Springer-Verlag, Berlin Heidelberg.
84. Furnish G. (May/June 1997) Disambiguated glommable expression templates. Compuers in Physics 11(3): 263-269.
85. George P. L. and Borouchaki H. (1998) Delaunay Triangulation and Meshing. Editions Hermes, Paris.
86. Gervasio P. (2008) Convergence analysis of high order algebraic fractional step schemes for timedependent stokes equations. SINUM.
87. Grepl M. A., Maday Y., Nguyen N. C., and Patera A. T. (2007) Efficient reducedbasis treatment of nonaffine and nonlinear partial differential equations. M2AN (Math. Model. Numer. Anal.).
88. Gill P., Murray W., Saunders M., and Wright M. (1989) Constrained nonlinear programming. Elsevier Handbooks In Operations Research AndManagement Science Optimization. Elsevier North-Holland, Inc., New York.
89. ndSandia Workshop of PDE-ConstrainedOptimization: Real-Time PDE-ConstrainedOptimization, SIAM Computational Science and Engineering Book Series, pages 197-216.
90. GreplM. A. and Patera A. T. (2005) A Posteriori error bounds for reduced-basis approximations of parametrized parabolic partial differential equations. M2AN (Math. Model. Numer. Anal.) 39(1): 157-181.
91. Gunzburger M. D., Peterson J., and Shadid J. N. (2007) Reduced-ordermodeling of time-dependent PDEs with multiple parameters in the boundary data. Comp. Meth. Applied Mech. 196: 1030-1047.
92. Girault V. and Raviart P. (1986) Finite Element Approximation of the Navier-Stokes Equations. Springer-Verlag, Berlin.
93. Godlewski E. and Raviart P. A. (1996) Hyperbolic Systems of Conservations Laws, volume 118. Springer-Verlag, New York.
94. Grepl M. (May 2005) Reduced-Basis Approximations and A Posteriori Error Estimation for Parabolic Partial Differential Equations. PhD thesis, Massachusetts Institute of Technology.
95. Grisvard P. (1976) Behaviour of the solutions of an elliptic boundary value problem in a polygonal or polyhedral domain. Numerical Solution of Partial Differential Equations, III. Academic Press, New York.
96. Grossmann C., Ross H., and Stynes M. (2007) Numerical treatment of Partial Differential Equations. Springer, Heidelberg, Heidelberg.
97. Gervasio P., Saleri F., and Veneziani A. (2006) Algebraic fractional-step schemes with spectral methods for the incompressible Navier-Stokes equations. J. Comput. Phys. 214(1): 347-365.
98. Gunzburger M. D. (1989) Finite Element Methods for Viscous Incompressible Flows. Academic Press, Boston.
99. Gunzburger M. D. (2003) Perspectives in Flow Control and Optimization. Advances in Design and Control. SIAM.
100. Gunzburger M. (2003) Perspectives in Flow Control and Optimization. Advances in Design and Control. SIAM, Philadelphia.
101. Hnat J. and Buckmaster J. (1976) Spherical cap bubbles and skirt formation. Phys. Fluids 19: 162-194.
102. Hirsh C. (1988) NumericalComputation of Internal and External Flows, volume 1. JohnWiley and Sons, Chichester.
103. Hirt C.W. and Nichols B. D. (1981) Volume of fluid (VOF) method for the dynamics of free boundaries. J. Comp. Phys. 39: 201-225.
104. Huynh D. B. P., Nguyen C. N., Rozza G., and Patera A. T. (2007-08) rbMIT Software: http://augustine.mit.edu/methodology/methodology_rbMIT_System.htm. © MIT, Tech. Lic. Office 12600, Cambridge, MA.
105. Hou T. Y. (1995) Numerical solutions to free boundary problems. ACTA Numerica 4: 335-415.
106. Huynh D. B. P., Rozza G., Sen S., and Patera A. T. (2007) A successive constraint linear optimization method for lower bounds of parametric coercivity and inf-sup stability constants. C. R. Acad. Sci. Paris, Analyse Numérique 345(8): 473-478.
107. Heywood J. G., Rannacher R., and Turek S. (1996) Artificial boundaries and flux and pressure conditions for the incompressible Navier-Stokes equations. Internat. J. Numer. Methods Fluids 22(5): 325-352.
108. Hughes T. J. R. (2000) The Finite Element Method. Linear Static and Dynamic Finite Element Analysis. Dover Publishers, New York.
109. Hamacher V. C., Vranesic Z. G., and Zaky S. G. (1997) Introduzione all'architettura dei calcolatori. McGraw Hill Italia, Milano.
110. Harlow F. H. and Welch J. E. (1965) Numerical calculation of time-dependent viscous incompressible flow of fluid with free surface. Physics of Fluids 8(12): 2182-2189.
111. Hou T. Y., Yang D. P., and Ran H. (2008) Multiscale analysis and computation for the 3d incompressible navier-stokes equations. SIAM Multiscale Modeling and Simulation 6 (4): 1317-1346.
112. Ito K. and Ravindran S. S. (1998) A reduced basis method for control problems governed by PDEs. In Desch W., Kappel F., and Kunisch K. (eds) Control and Estimation of Distributed Parameter Systems, pages 153-168. Birkhäuser.
113. Ito K. and Ravindran S. S. (1998) A reduced-ordermethod for simulation and control of fluid flows. Journal of Computational Physics 143(2): 403-425.
114. Ito K. and Ravindran S. S. (2001) Reduced basis method for optimal control of unsteady viscous flows. International Journal of Computational Fluid Dynamics 15(2): 97-113.
115. Infante J. and Zuazua E. (1999) Boundary observability for the space semi-discretizations of the 1-d wave equation. M2AN Math. Model. Numer. Anal. 33(2): 407-438.
116. Jameson A. (1988) Optimum aerodynamic design using CFD and control theory. AIAA Paper 95-1729-CP pages 233-260.
117. JoergM. (2005)Numerical investigations ofwall boundary conditions for two-fluid flows. Master's thesis, École Polytechnique Fédérale de Lausanne.
118. Johnson C. (1987) Numerical Solution of Partial Differential Equations by the Finite ElementMethod. Cambridge University Press, Cambridge.
119. Knabner P. and Angermann L. (2000) Numerical Methods for Elliptic and Parabolic Partial Differential Equations, volume 44 of TAM. Springer-Verlag, New York.
120. Kim S., Alonso J., and Jameson A. (2002) Design optimization of hight-lift configurations using a viscous continuos adjoint method. AIAA paper, 40th AIAA Aerospace SciencesMeeting and Exibit, Jan 14-17 2002 0844.
121. Kolmogorov A. and Fomin S. (1989) Elements of the Theory of Functions and Functional Analysis. V.M. Tikhominov, Nauka-Moscow.
122. Kajitani H., Miyata H., Ikehata M., Tanaka H., Adachi H., Namimatzu M., and Ogiwara S. (1983) Summary of the cooperative experiment on Wigley parabolic model in Japan. In Proc. of the 2nd DTNSRDC Workshop on Ship Wave Resistance Computations (Bethesda, USA), pages 5-35.
123. Kawohl B., Pironneau O., Tartar L., and Zolesio J. (2000) Optimal Shape Design. Springer-Verlag, Berlin.
124. Kroener D. (1997) Numerical Schemes for Conservation Laws. Wiley-Teubner, Chichester.
125. Karniadakis G. E. and Sherwin S. J. (2005) Spectral/hp ElementMethods for Computational Fluid Dynamics. Oxford University Press, New York, II edition.
126. Kunisch K. and Volkwein S. (2002) Galerkin proper orthogonal decomposition methods for a general equation in fluid dynamics. SIAM J. Num. Analysis 40(2): 492-515.
127. LeGresley P. A. and Alonso J. J. (2000) Airfoil design optimization using reduced order models based on proper orthogonal decomposition. In Fluids 2000 Conference and Exhibit, Denver, CO. Paper 2000-2545.
128. Le Bris C. (2005) Systèmes multiiéchelles: modélisation et simulation, volume 47 of Mathématiques et Applications. Springer, Paris.
129. Lee M. Y. L. (1991) Estimation of the error in the reduced-basismethod solution of differential algebraic equations. SIAM Journal of Numerical Analysis 28: 512-528.
130. LeVeque R. J. (2002) Finite VolumeMethods for Hyperbolic Problems. Cambridge Texts in AppliedMathematics.
131. LeVeque R. J. (2002) NumericalMethods for Conservation Laws. Birkhäuser Verlag, Basel, II edition.
132. LeVeque R. J. (2007) Finite DifferenceMethods for Ordinary and Partial Differential Equations: Steady-State and Time-Dependent Problems. SIAM, Philadelphia.
133. Lions J. (1971) Optimal Control of Systems Governed by Partial Differential Equations. Springer-Verlag, New York.
134. Lions J. (1972) Some Aspects of the Optimal Control of Distribuited Parameter Systems. SIAM, Philadelphia.
135. Lions P.-L. (1996) Mathematical topics in fluid mechanics. Vol. 1, volume 3 of Oxford Lecture Series in Mathematics and its Applications. The Clarendon Press Oxford University Press, New York. Incompressible models, Oxford Science Publications.
136. Landau L. D. and Lifshitz E. M. (1959) Fluid mechanics. Translated from the Russian by J. B. Sykes and W. H. Reid. Course of Theoretical Physics, Vol. 6. Pergamon Press, London.
137. Lippman S. B. and Lajoie J. (2000) C++ Corso di Programmazione. Addison Wesley Longman Italia, Milano, III edition.
138. Lions J. L. and Magenes E. (1968) Quelques Méthodes des Résolution des Problémes aux Limites non Linéaires. Dunod, Paris.
139. Loeve M. M. (1955) Probability Theory. Van Nostrand, Toronto-New York-London.
140. Lin S. P. and Reitz R. D. (1998) Drop and spray formation from a liquid jet. Annu. Rev. Fluid Mech. 30: 85-105.
141. 2norms. Comp. Struct. 53: 907-919.
142. Marchuk G. I. (1995) Adjoint Equations and Analysis of Complex Systems. Kluwer Academic Publishers, Dordrecht.
143. Maurer H. (1981) First and second order sufficient optimality conditions in mathematical programming and optimal control. Mathematical Programming Study 14: 163-177.
144. Maxworthy T. (1976) Experiments on collisions between solitary waves. Journal of Fluid Mechanics 76: 177-185.
145. Meyer C. D. (2000) Matrix Analysis and Applied Linear Algebra. SIAM.
146. Mulder W., Osher S., and Sethian J. (1992) Computing interface motion in compressible gas dynamics. Journal of Computational Physics 100(2): 209-228.
147. Mohammadi B. and Pironneau O. (1994) Analysis of the K-Epsilon Turbulence Model. JohnWiley & Sons, Chichester.
148. Muzaferija S. and Peric M. (1997) Computation of free-surface flows using finite volume method and moving grids. Numer. Heat Trans., Part B 32: 369-384.
149. Mohammadi B. and Pironneau O. (2001) Applied Shape Optimization for Fluids. Clarendon Press, Oxford.
150. Maday Y., Patera A. T., and Turinici G. (2002) Global a priori convergence theory for reduced-basis approximation of single-parameter symmetric coercive elliptic partial differential equations. C. R. Acad. Sci. Paris, Série I 335(3): 289-294.
151. Maday Y., Patera A., and Turinici G. (2002) A Priori convergence theory for reduced-basis approximations of single-parameter elliptic partial differential equations. Journal of Scientific Computing 17(1-4): 437-446.
152. Noor A. K., Balch C. D., and Shibut M.A. (1984) Reductionmethods for non-linear steady-state thermal analysis. Int. J. Num. Meth. Engrg. 20: 1323-1348.
153. Nitsche J. A. (1968) Ein kriterium für die quasi-optimalitat des Ritzchen Verfahrens. Numer. Math. 11: 346-348.
154. Noor A. K. (1981) Recent advances in reduction methods for nonlinear problems. Comput. Struct. 13: 31-44.
155. Noor A. K. (1982) On making large nonlinear problems small. Comp. Meth. Appl. Mech. Engrg. 34: 955-985.
156. Noor A. K. and Peters J. M. (1980) Reduced basis technique for nonlinear analysis of structures. AIAA Journal 18(4): 455-462.
157. Nguyen N. C., Rozza G., Huynh D. B. P., and Patera A. T. (2008) Reduced basis approximation and a posteriori error estimation for parametrized parabolic pdes; Application to real-time Bayesian parameter estimation. In Biegler L., Biros G., Ghattas O., Heinkenschloss M., Keyes D., Mallick B., Tenorio L., van Bloemen Waanders B., andWillcox K. (eds) ComputationalMethods for Large Scale InverseProblems and Uncertainty Quantification. JohnWiley and Sons, Chichester.
158. Nguyen N. C., Veroy K., and Patera A. T. (2005) Certified real-time solution of parametrized partial differential equations. In Yip S. (ed) Handbook of Materials Modeling, pages 1523-1558. Springer, Netherlands.
159. Nocedal J. and Wright S. (2006) Numerical Optimization. Springer Series in Operations Research and Financial Engineering. Springer, New York.
160. Naga A. and Zhang Z. (2004) A posteriori error estimates based on the polynomial preserving recovery. SIAM J. Numer. Anal. 42: 1780-1800.
161. Oliveira I. and Patera A. (2007) Reduced-basis techniques for rapid reliable optimization of systems described by affinely parametrized coercive elliptic partial differential equations. Optimization and Engineering 8(1): 43-65.
162. Osher S. and Sethian J. A. (1988) Fronts propagating with curvature-dependent speed: algorithms based on Hamilton-Jacobi formulations. J. Comput. Phys. 79(1): 12-49.
163. Patankar S. V. (1980) Numerical Heat Transfer and Fluid Flow. Hemisphere, Washington.
164. Patera A. (1984) A spectral element method for fluid dynamics: laminar flow in a channel expansion. J. Comput. Phys. 54: 468-488.
165. www.mcs.anl.gov/petsc/.
166. Peterson J. S. (1989) The reduced basis method for incompressible viscous flow calculations. SIAM J. Sci. Stat. Comput. 10(4): 777-786.
167. Pironneau O. (1984) Optimal Shape Design for Elliptic Systems. Springer-Verlag, New York.
168. Porsching T. A. and Lee M. Y. L. (1987) The reduced-basismethod for initial value problems. SIAM Journal of Numerical Analysis 24: 1277-1287.
169. Porsching T. A. (1985) Estimation of the error in the reduced basis method solution of nonlinear equations. Mathematics of Computation 45(172): 487-496.
170. Parolini N. and Quarteroni A. (2005) Mathematical models and numerical simulations for the america's cup. Comput. Methods Appl. Mech. Engrg. 194(9-11): 1001-1026.
171. Parolini N. and Quarteroni A. (2007)Modelling and numerical simulation for yacht engineering. In Proceedings of the 26th Symposium on Naval Hydrodynamics. Strategic Analysis, Inc., Arlington, VA.
172. Patera A. T. and Rozza G. (2006-2007) Reduced Basis Approximation and A Posteriori Error Estimation for Parametrized Partial Differential Equations. Copyright MIT. To appear in MIT PappalardoMonographs in Mechanical Engineering.
173. Prohl A. (1997) Projection and Quasi-Compressibility Methods for Solving the IncompressibleNavier-Stokes Equations. Advances in NumericalMathematics. B.G. Teubner, Stuttgart.
174. Prud'homme C. (2006)A domain specificembedded language in c++ for automatic differentiation, projection, integration and variational formulations. Scientific Programming 14(2): 81-110.
175. Prud'homme C., Rovas D., Veroy K., Maday Y., Patera A., and Turinici G. (2002) Reliable real-time solution of parametrized partial differential equations: Reducedbasis output bounds methods. Journal of Fluids Engineering 124(1): 70-80.
176. Pawlak T. P., Wheeler M. J., and Yunus S. M. (1990) Application of the Zienkiewicz-Zhu error estimator for plate and shell analysis. Int. J. Numer. Methods Eng. 29: 1281-1298.
177. Quarteroni A. and Rozza G. (2007) Numerical solution of parametrized Navier-Stokes equations by reduced basis method. Num. Meth. PDEs 23: 923-948.
178. Quarteroni A., Rozza G., Dedè L., and Quaini A. (2006) Numerical approximation of a control problem for advection-diffusion processes. System modeling and optimization. IFIP Int. Fed. Inf. Process. 199: 261-273.
179. Quarteroni A., Rozza G., and Quaini A. (2006) Reduced basis method for optimal control af advection-diffusion processes. In Fitzgibbon W., Hoppe R., Periaux J., Pironneau O., and VassilevskiY. (eds) Advances in NumericalMathematics, pages 193-216. Moscow, Institute of Numerical Mathematics, Russian Academy of Sciences and Houston, Department of Mathematics, University of Houston.
180. Quarteroni A., Sacco R., and Saleri F. (2007) Numerical Mathematics. Springer, Berlin Heidelberg, II edition.
181. Quarteroni A., Saleri F., and Veneziani A. (2000) Factorization methods for the numerical approximation of Navier-Stokes equations. Comput. Methods Appl. Mech. Engrg. 188(1-3): 505-526.
182. Qu Z. (2002) Unsteady open-channel flow over a mobile bed. PhD thesis, École Polytechnique Fédérale de Lausanne.
183. Quartapelle L. (1993) Numerical Solution of the Incompressible Navier-Stokes Equations. Birkhäuser Verlag, Basel.
184. Quarteroni A. and Valli A. (1994) Numerical Approximation of Partial Differential Equations. Springer, Berlin Heidelberg.
185. Quarteroni A. and Valli A. (1999) Domain DecompositionMethods for Partial Differential Equations. Oxford Science Publications, Oxford.
186. Rannacher R. (1999) Error control in finite element computations. An introduction to error estimation and mesh-size adaptation. In Error control and adaptivity in scientific computing (Antalya, 1998), pages 247-278. Kluwer Acad. Publ., Dordrecht.
187. Ravindran S. S. (2000) A reduced order approach to optimal control of fluids flow using proper orthogonal decomposition. Int. J. of Numerical Methods in Fluids 34(5): 425-448.
188. Rhie C. M. and Chow W. L. (1983) Numerical study of the turbulent flow past an airfoil with trailing edge separation. AIAA Journal 21(11): 1525-1532.
189. RheinboldtW. C. (1981) Numerical analysis of continuation methods for nonlinear structural problems. Computers and Structures 13(1-3): 103-113.
190. Rheinboldt W. C. (1993) On the theory and error estimation of the reduced basis method for multi-parameter problems. Nonlinear Analysis, Theory, Methods and Applications 21(11): 849-858.
191. Rozza G., Huynh D. B. P., and Patera A. T. (2008) Reduced basis approximation and a posteriori error estimation for affinely parametrized elliptic coercive partial differential equations:Application to transport and continuummechanics. Archives Computational Methods in Engineering 15(3): 229-275.
192. Rodríguez R. (1994) Some remarks on Zienkiewicz-Zhu estimator. Numer. Methods Part. Diff. Eq. 10: 625-635.
193. Rozza G. (2005) Real-time reduced basis techniques for arterial bypass geometries. In Bathe K. (ed) Computational Fluid and Solid Mechanics, pages 1283-1287. Elsevier. Proceedings of the Third M.I.T. Conference on Computational Fluid and Solid Mechanics, June 14-17, 2005.
194. Rozza G. (2005) Reduced-basismethods for elliptic equations in sub-domainswith a posteriori error bounds and adaptivity. Appl. Numer. Math. 55(4): 403-424.
195. Rozza G. (2009) Reduced basis method for Stokes equations in domains with nonaffine parametric dependence. Comp. Vis. Science 12: 23-35.
196. Rozza G. and Patera A. T. (2008) Worked Problems with rbMIT Software: http://augustine.mit.edu/workedProblems.htm. ©MIT, Cambridge, MA.
197. RenardyM. and Rogers R. C. (2004) An Introduction to Partial Differential Equations. Springer-Verlag, New York, II edition.
198. Ross H. G., Stynes M., and Tobiska L. (1996) Numerical Methods for Singularly Perturbed Differential Equations. Convection-Diffusion and Flow Problems. Springer-Verlag, Berlin Heidelberg.
199. Rudin W. (1991) Analyse Rèelle et Complexe. Masson, Paris.
200. Rozza G. and Veroy K. (2007) On the stability of reduced basis method for Stokes equations in parametrized domains. Comp. Meth. Appl. Mech. and Eng. 196: 1244-1260.
201. SaadY. (1996) IterativeMethods for Sparse Linear Systems. PWS Publishing Company, Boston.
202. Sagaut P. (2006) LargeEddy Simulation for IncompressibleFlows: an Introduction. Springer-Verlag, Berlin Heidelberg, III edition.
203. Salsa S. (2008) Partial Differential Equations in Action-From Modelling to Theory. Springer, Milan.
204. Schwarz H. (1869) Über einige abbildungsdufgaben. J. Reine Agew. Math. 70: 105-120.
205. Schwab C. (1998) p and hp-Finite ElementMethods. Oxford Science Publication, Oxford.
206. Strang G. and Fix G. J. (1973) An Analysis of the Finite ElementMethod. Wellesley-Cambridge Press, Wellesley, MA.
207. Shewchuk J. R. www.cs.cmu.edu/ quake/triangle.html.
208. Sirovich L. (1987) Turbulence and the dynamics of coherent structures, part 1: Coherent structures. Quarterly of Applied Mathematics 45(3): 561-571.
209. Smolianski A. (2001) Numerical Modeling of Two-Fluid Interfacial Flows. PhD thesis, University of Jyväskylä.
210. Spivak M. (1999) A comprehensive introduction to differential geometry. Vol. II. Publish or Perish Inc., Houston, Tex., III edition.
211. Stroud A. H. (1971) Approximate calculation of multiple integrals. Prentice-Hall, Inc., Englewood Cliffs, N.J.
212. Strickwerda J. C. (1989) Finite Difference Schemes and Partial Differential Equations. Wadworth & Brooks/Cole, Pacific Grove.
213. Strostroup B. (2000) C++ Linguaggio, Libreria Standard, Principi di Programmazione. AddisonWelsey Longman Italia, Milano, III edition.
214. Saleri F. and Veneziani A. (2005) Pressure correction algebraic splitting methods for the incompressible Navier-Stokes equations. SIAM J. Numer. Anal. 43(1): 174-194.
215. Sokolowski J. and Zolesio J. (1991) Introduction to Shape Optimization (Shape Sensitivity Analysis). Springer-Verlag, New York.
216. Tanaka N. (1993) Global existence of two phase nonhomogeneous viscous incompressible fluid flow. Comm. Partial Differential Equations 18(1-2): 41-81.
217. Trefethen L. and Embree M. (2005) Spectra and pseudospectra. The behavior of nonnormal matrices and operators. Princeton University Press, Princeton.
218. Temam R. (2001) Navier Stokes Equations. North-Holland, Amsterdam.
219. Tsuchiya K. and Fan L.-S. (1988) Near-wake structure of a single gas bubble in a two-dimensional liquid-solid fluidized bed: vortex shedding and wake size variation. Chem. Engrg. Sci. 43(5): 1167-1181.
220. Thomee V. (1984) Galerkin Finite Element Methods for Parabolic Problems. Springer, Berlin and Heidelberg.
221. Trefethen L. and III D. B. (1997) Numerical Linear Algebra. SIAM.
222. Taylor A. and Lay D. (1958) Introduction to Functional Analysis. J. Wiley & Sons, New York.
223. Toro E. (1999) Riemann Solvers and Numerical Methods for Fluid Dynamics. Springer-Verlag, Berlin.
224. software. sandia. gov/trilinos/.
225. Thompson J. F., Soni B. K., and Weatherill N. P. (eds) (1999) Handook of Grid Generation. CRC Press, Boca Raton.
226. Toselli A. and Widlund O. (2005) Domain Decomposition Methods-Algorithms and Theory. Springer-Verlag, Berlin Heidelberg.
227. Thompson J. F., Warsi Z. U. A., andMastin C.W. (1985) Numerical Grid Generation, Foundations and Applications. North-Holland, New York.
228. www.cise.ufl.edu/research/sparse/umfpack/.
229. ur Rehman M., Vuik C., and Segal G. (2008) A comparison of preconditioners for incompressible navier-stokes solvers. Int. J. Numer. Meth. Fluids 57: 1731-1751.
230. ur Rehman M., Vuik C., and Segal G. (2009) Preconditioners for the steady incompressible navier-stokes problem. Int. J. Applied Math. 38 (4).
231. Vasiliev F. (1981) Methods for Solving the Extremum Problems. Nauka, Moscow.
232. van der Vorst H. A. (2003) Iterative Krylov Methods for Large Linear Systems. Cambridge University Press, Cambridge.
233. Veldhuizen T. (1995) Expression templates. C++ Report Magazine 7(5): 26-31. see also the web page http://osl.iu.edu/ tveldhui.
234. Veneziani A. (1998) Mathematical and Numerical Modeling of Blood Flow Problems. PhD thesis, Università degli Studi di Milano.
235. Verfürth R. (1984) Error estimates for a mixed finite elements approximation of the stokes equations. R.A.I.R.O. Anal. Numér. 18: 175-182.
236. Verführth R. (1996) A Review of a Posteriori Error Estimation and Adaptive Mesh Refinement Techniques. Wiley-Teubner, New York.
237. Versteeg H. and Malalasekra W. (1996) An Introduction to Computational Fluid Dynamics: the Finite Volume Method Approach. Prentice-Hall.
238. Veroy K. and Patera A. T. (2005) Certified real-time solution of the parametrized steady incompressible Navier-Stokes equations; Rigorous reduced-basis a posteriori error bounds. International Journal for Numerical Methods in Fluids 47: 773-788.
239. Veroy K., Prud'homme C., and Patera A. T. (2003) Reduced-basis approximation of the viscous Burgers equation: Rigorous a posteriori error bounds. C. R. Acad. Sci. Paris, Série I 337(9): 619-624.
240. Veroy K., Prud'homme C., Rovas D. V., and Patera A. T. (2003) A Posteriori error bounds for reduced-basis approximation of parametrized noncoercive and nonlinear elliptic partial differential equations. In Proceedings of the 16th AIAA Computational Fluid Dynamics Conference. Paper 2003-3847.
241. Wesseling P. (2001) Principles of Computational Fluid Dynamics. Springer-Verlag, Berlin Heidelberg New York.
242. Wilcox D. C. (1998) Turbulence Modeling in CFD. DCW Industries, La Cañada, CA, II edition.
243. Winkelmann C. (2007) Interior penalty finite element approximation of Navier-Stokes equations and application to free surface flows. PhD thesis, École Polytechnique Fédérale de Lausanne.
244. Willcox K. and Peraire J. (2002) Balanced model reduction via the proper orthogonal decomposition. AIAA Journal 40(11): 2323-2330.
245. Wyatt D. C. (2000) Development and assessment of a nonlinear wave prediction methodology for surface vessels. Journal of ship research 44: 96.
246. Yosida K. (1974) Functional Analysis. Springer-Verlag, Berlin Heidelberg.
247. Zienkiewicz O. (2000) Achievements and some unsolved problems of the finite element method. Int. J. Numer. Meth. Eng. 47: 9-28.
248. Zienkiewicz O. C. and Taylor R. L. (2000) The Finite Element Method, Vol. 1, The Basis. Butterworth-Heinemann, Oxford, V edition.
249. Zuazua E. (2003) Propagation, observation, control and numerical approximation of waves. Bol. Soc. Esp. Mat. Apl. 25: 55-126.
250. Zuazua E. (2005) Propagation, observation, and control of waves approximated by finite difference methods. SIAM Review 47 (2): 197-243.
251. Zuazua E. (2006) Controllability and observability of partial differential equations: Some results and open problems. In Dafermos C. and Feiteisl E. (eds) Handbook of Differential Equations: Evolutionary Differential Equations, volume 3, pages 527-621. Elsevier Science.
252. Zienkiewicz O. C. and Zhu J. Z. (1992) The superconvergent patch recovery and a posteriori error estimates. I: The recovery technique. Int. J. Numer. Meth. Engng. 33: 1331-1364.