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Improved successive constraint method based a posteriori error estimate for reduced basis approximation of 2D Maxwell's problem

Published online by Cambridge University Press:  21 August 2009

Yanlai Chen ,
Jan S. Hesthaven ,
Yvon Maday  and
Jerónimo Rodríguez
Yanlai Chen
Affiliation:
Division of Applied Mathematics, Brown University, 182 George St, Providence, RI 02912, USA. Yanlai_Chen@brown.edu; Jan.Hesthaven@Brown.edu
Jan S. Hesthaven
Affiliation:
Division of Applied Mathematics, Brown University, 182 George St, Providence, RI 02912, USA. Yanlai_Chen@brown.edu; Jan.Hesthaven@Brown.edu
Yvon Maday
Affiliation:
Division of Applied Mathematics, Brown University, 182 George St, Providence, RI 02912, USA. Yanlai_Chen@brown.edu; Jan.Hesthaven@Brown.edu Université Pierre et Marie Curie-Paris 6, UMR 7598, Laboratoire J.-L. Lions, 75005 Paris, France. maday@ann.jussieu.fr
Jerónimo Rodríguez
Affiliation:
Departamento de Matemática Aplicada, Universidade de Santiago de Compostela, 15782 Santiago de Compostela, Spain. jeronimo.rodriguez@usc.es
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Abstract


In a posteriori error analysis of reduced basis approximations to affinely parametrized partial differential equations, the construction of lower bounds for the coercivity and inf-sup stability constants is essential. In [Huynh et al., C. R. Acad. Sci. Paris Ser. I Math.345 (2007) 473–478], the authors presented an efficient method, compatible with an off-line/on-line strategy, where the on-line computation is reduced to minimizing a linear functional under a few linear constraints. These constraints depend on nested sets of parameters obtained iteratively using a greedy algorithm. We improve here this method so that it becomes more efficient and robust due to two related properties: (i) the lower bound is obtained by a monotonic process with respect to the size of the nested sets; (ii) less eigen-problems need to be solved. This improved evaluation of the inf-sup constant is then used to consider a reduced basis approximation of a parameter dependent electromagnetic cavity problem both for the greedy construction of the elements of the basis and the subsequent validation of the reduced basis approximation. The problem we consider has resonance features for some choices of the parameters that are well captured by the methodology.


Keywords

Reduced basis methodsuccessive constraint methodinf-sup constanta posteriori error estimateMaxwell's equationdiscontinuous Galerkin method.
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 43 , Issue 6 , November 2009 , pp. 1099 - 1116
Copyright
© EDP Sciences, SMAI, 2009

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References

Barrault, M., Nguyen, N.C., Maday, Y. and Patera, A.T.. An empirical interpolation method: Application to efficient reduced-basis discretization of partial differential equations. C. R. Acad. Sci. Paris Ser. I Math. 339 (2004) 667–672. CrossRef
Barret, A. and Reddien, G., On the reduced basis method. Z. Angew. Math. Mech. 75 (1995) 543–549. CrossRef
Bui-Thanh, T., Willcox, K. and Ghattas, O., Model reduction for large-scale systems with high-dimensional parametric input space. SIAM J. Sci. Comput. 30 (2008) 3270–3288. CrossRef
Chen, Y., Hesthaven, J.S., Maday, Y. and Rodríguez, J., A monotonic evaluation of lower bounds for Inf-Sup stability constants in the frame of reduced basis approximations. C. R. Acad. Sci. Paris Ser. I Math. 346 (2008) 1295–1300. CrossRef
Grepl, M.A., Maday, Y., Nguyen, N.C. and Patera, A.T.. Efficient reduced-basis treatment of nonaffine and nonlinear partial differential equations. ESAIM: M2AN 41 (2007) 575–605. CrossRef
M.D. Gunzburger, Finite element methods for viscous incompressible flows. Academic Press (1989).
J.S. Hesthaven and T. Warburton, Nodal Discontinuous Galerkin Methods: Algorithms, Analysis, and Applications, Springer Texts in Applied Mathematics 54. Springer Verlag, New York (2008).
Huynh, D.B.P., Rozza, G., Sen, S. and Patera, A.T., A successive constraint linear optimization method for lower bounds of parametric coercivity and inf-sup stability constants. C. R. Acad. Sci. Paris Ser. I Math. 345 (2007) 473–478. CrossRef
Machiels, L., Maday, Y., Oliveira, I.B., Patera, A.T. and Rovas, D., Output bounds for reduced-basis approximations of symmetric positive definite eigenvalue problems. C. R. Acad. Sci. Paris Ser. I Math. 331 (2000) 153–158. CrossRef
Y Maday, Reduced Basis Method for the Rapid and Reliable Solution of Partial Differential Equations, in Proceeding ICM Madrid (2006).
Y. Maday, A.T. Patera and D.V. Rovas, A blackbox reduced-basis output bound method for noncoercive linear problems, in Nonlinear Partial Differential Equations and Their Applications, D. Cioranescu and J.L. Lions Eds., Collège de France Seminar XIV, Elsevier Science B.V. (2002) 533–569.
Nagy, D.A., Modal representation of geometrically nonlinear behaviour by the finite element method. Comput. Struct. 10 (1979) 683–688. CrossRef
N.C. Nguyen, K. Veroy and A.T. Patera. Certified real-time solution of parametrized partial differential equations, in Handbook of Materials Modeling, S. Yip Ed., Springer (2005) 1523–1558.
Noor, A.K. and Peters, J.M., Reduced basis technique for nonlinear analysis of structures. AIAA Journal 18 (1980) 455–462.
Peterson, J.S., The reduced basis method for incompressible viscous flow calculations. SIAM J. Sci. Stat. Comput. 10 (1989) 777–786. CrossRef
Prud'homme, C., Rovas, D., Veroy, K., Maday, Y., Patera, A.T. and Turinici, G., Reliable realtime solution of parametrized partial differential equations: Reduced-basis output bound methods. J. Fluids Engineering 124 (2002) 70–80. CrossRef
Rozza, G., Huynh, D.B.P. and Patera, A.T., Reduced basis approximation and a posteriori error estimation for affinely parametrized elliptic coercive partial differential equations: Application to transport and continuum mechanics. Arch. Comput. Methods Eng. 15 (2008) 229–275. CrossRef
Sen, S., Veroy, K., Huynh, D.B.P., Deparis, S., Nguyen, N.C. and Patera, A.T., “Natural norm” a posteriori error estimators for reduced basis approximations. J. Comput. Phys. 217 (2006) 37–62. CrossRef