Reduced basis method for parameterized elliptic advection-reaction problems

Journal of Computational Mathematics

Volumn 28, Issue 1, 2010, Pages 122-148

  Dedè, Luca ^a  

^a POLITECNICO DI MILANO   (Italy)

Author keywords

"Primal dual"Reduced basis approach; A posteriori error estimation; Parametrized advection reaction partial differential equations; Reduced Basis method; Stabilized finite element method

Indexed keywords

EID: 77749320504     PISSN: 02549409     EISSN: None     Source Type: Journal    
DOI: 10.4208/jcm.2009.09-m3015     Document Type: Article

References (31)

1. B.O. Almroth, P. Stern, F.A. Brogan, Automatic choice of global shape functions in structural analysis, AIAA J., 16 (1978), 525-528.
2. I. Babǔiska, Error-bounds for finite element method, Numer. Math., 16 (1971), 322-333.
3. É. Cancès, C. Le Bris, Y. Maday, N.C. Nguyen, A.T. Patera, G.S.H. Pau, Feasibility and competitiveness of a reduced basis approach for rapid electronic structure calculations in quantum chemistry, High dimensional partial differential equations in science and engineering, CRM, Proceedings and Lecture Notes, 41, Amer. Math. Soc, Providence RI, 2007, 15-47.
4. L. Dedè, Adaptive and reduced basis methods for optimal control problems in environmental applications, PhD Thesis, Politecnico di Milano, 2008. http://mox.polimi.it
5. L. Dedè, A. Quarteroni, Optimal control and numerical adaptivity for advection-diffusion equations, Math. Model. Numer. Anal, 39:5 (2005), 1019-1040.
6. M.A. Grepl, A.T. Patera, A posteriori error bounds for reduced-basis approximations of parametrized parabolic partial differential equations, Math. Model. Numer. Anal., 39:1 (2005), 157-181.
7. D.B.P. Huynh, A.T. Patera, Reduced basis approximation and a posteriori error estimation fro stress intensity factors, Int. J. Numer. Meth. Eng., 72:10 (2007), 1219-1259.
8. C. Johnson, Numerical Solution of Partial Differential Equations by the Finite Element Method, Cambridge University Press, Cambridge, 1987.
9. C. Johnson, A.H. Schatz, L.B. Wahlbin, Crosswind smear and pointwise errors in streamline diffusion finite element methods, Math. Comput., 49:179 (1987), 25-38.
10. J.L. Lions, E. Magenes, Non-homogeneous Boundary Value Problems and Applications, Vol.1., Springer-Verlag, New York and Heidelberg, 1972.
11. A.E. Løvgren, Y. Maday, E. Rønquist, A reduced basis element method for the steady Stokes problem, Math. Model. Numer. Anal., 40:3 (2006), 529-552.
12. R. Milani, A. Quarteroni, G. Rozza, Reduced basis method for linear elasticity problems with many parameters, Comput. Method. Appl. M., 197:51-52 (2008), 4812-4829.
13. N.C. Nguyen, K. Veroy, A.T. Patera, Certified real-time solution of parametrized partial differential equations, in Handbook of Materials Modeling, S. Yip (ed.), Springer, 2005, 1523-1558.
14. A.K. Noor, J.M. Peters, Reduced basis technique for nonlinear analysis of structures, AIAA J., 19 (1980), 455-462.
15. A.T. Patera, E. Rønquist, Reduced basis approximations and a posteriori error estimation for a Boltzmann model, Comput. Method. Appl. M., 196:29-30 (2007), 2925-2942.
16. A.T. Patera, G. Rozza, Reduced Basis Approximation and A Posteriori Error Estimation for Parametrized Partial Differential Equations, Version 1.0, Copyright MIT 2006-08, to appear in (tentative rubric) MIT Pappalardo Graduate Monographs in Mechanical Engineering, http://augustine.mit.edu/
17. T.A. Porsching, Estimation of the error in the reduced basis method solution of nonlinear equations, Math. Comput., 45:172 (1985), 487-496.
18. C. Prud'homme, D. Rovas, K. Veroy, Y. Maday, A.T. Patera, G. Turinici, Reliable real-time solution of parametrized partial differential equations: reduced-basis output bound methods, J. Fluids Eng., 124:1 (2002), 70-80.
19. A. Quarteroni, A. Quaini, G. Rozza, Reduced basis methods for advection-diffusion optimal control problems, in Advances in Numerical Mathematics, W. Fitzgibbon, R. Hoppe, J. Periaux, O. Pironneau and Y. Vassilevski (ed.s), Moscow, Russian Acedemy of Science and Houston, University of Houston, 2006, 193-216.
20. A. Quarteroni, G. Rozza, Numerical solutions of parametrized Navier-Stokes equations by reduced basis method, Numer. Meth. Part. Diff. Eqn., 23:4 (2007), 923-948.
21. A. Quarteroni, G. Rozza, L. Dede, A. Quaini, Numerical approximation of a control problem for advection-diffusion processes, in System modeling and optimization, IFIP Int. Fed. Inf. Process., 199, Springer, New York, 2006, pp. 261-273.
22. A. Quarteroni, R. Sacco, F. Saleri, Numerical Mathematics, Springer-Verlag, Berlin, 2007.
23. A. Quarteroni, A. Valli, Numerical Approximation of Partial Differential Equations, Springer, Berlin and Heidelberg, 1994.
24. G. Rozza, Reduced-basis methods for elliptic equations in sub-domains with a posteriori error bounds and adaptivity, Appl. Numer. Math., 55:4 (2005), 403-424.
25. G. Rozza, Reduced basis methods for Stokes equations in domains with non affine parameter dependence, Comput. Vis. Sci., 12:1 (2009), 23-35.
26. G. Rozza, D.B.P. Huynh, A.T. Patera, Reduced basis approximation and a posteriori error estimation for affinely parametrized elliptic coercive partial differential equations: application to transport and continuum mechanics, Arch. Comput. Method. E., 15:3 (2008), 229-275.
27. G. Rozza, K. Veroy, On the stability of the reduced basis method for Stokes equations in parametrized domains, Comput. Method. Appl. M., 196:7 (2007), 1244-1260.
28. S. Sen, K. Veroy, D.B.P. Huynh, S. Deparis, N.C. Nguyen, A.T. Patera, "Natural norm" a posteriori error estimators for reduced basis approximations, J. Comput. Phys., 217:1 (2006), 37-62.
29. A. Tan Yong Kwang, Reduced Basis Method for 2nd Order Wave Equation: Application to One-Dimension Seismic Problem, Master Thesis, Singapore-MIT Alliance, National University of Singapore, 2006. http://augustine.mit.edu/
30. K. Veroy, A.T. Patera, Certified real-time solution of the parametrized steady incompressible Navier-Stokes equations: rigorous reduced-basis a posteriori error bounds, Int. J. Numer. Method. Fl., 47 (2005), 773-788.
31. G. Zhou. How accurate is the streamline diffusion finite element method? Math. Comput., 66:217 (1997), 31-44.