Error bounds in high-order Sobolev norms for POD expansions of parameterized transient temperatures. (Estimations d’erreur d’ordre élevé pour la décomposition POD appliquée à l’équation de la chaleur parametrisée.) (English. Abridged French version) Zbl 1362.65095
Summary: In this work, we analyze the convergence of the proper orthogonal decomposition (POD) expansion for the solution to the heat conduction parameterized with respect to the thermal conductivity coefficient. We obtain error bounds for the POD approximation in high-order norms in space that assure an exponential rate of convergence, uniformly with respect to the parameter whenever it remains within a compact set of positive numbers. We present some numerical tests that confirm this theoretical accuracy.
MSC:
| 65M15 | Error bounds for initial value and initial-boundary value problems involving PDEs |
| 65M12 | Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs |
| 65M60 | Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs |
| 35K05 | Heat equation |
References:
| [1] | Azaïez, M.; Ben Belgacem, F.; Chacón Rebollo, T., Error bounds for POD expansions of parameterized transient temperatures, Comput. Methods Appl. Mech. Eng., 305, 501-511 (2016) · Zbl 1425.74446 |
| [2] | Brézis, H., Análisis Funcional. Teoría y aplicaciones (1984), Alianza Editorial: Alianza Editorial Madrid, Spain |
| [3] | Buffa, A.; Maday, Y.; Patera, A. T.; Prud homme, C.; Turicini, G., A priori convergence of the greedy algorithm for the parametrized reduced basis, Math. Model. Numer. Anal., 46, 595-603 (2012) · Zbl 1272.65084 |
| [4] | Cohen, A.; Devore, R., Approximation of high-dimensional parametric PDEs, Acta Numer., 1-159 (2015) · Zbl 1320.65016 |
| [5] | Cordier, L.; Bergmann, M., Proper Orthogonal Decomposition: An Overview. Lecture Series 2002-04 and 2003-04 on Post-Processing of Experimental and Numerical Data (2003), Von Karman Institute for Fluid Dynamics: Von Karman Institute for Fluid Dynamics Rhode-Saint-Genèse, Belgium |
| [6] | Hecht, F., New development in FreeFem++, J. Numer. Math., 20, 3-4, 251-265 (2012) · Zbl 1266.68090 |
| [7] | Hesthaven, J. S.; Rozza, G.; Stamm, B., Certified Reduced Basis Methods for Parametrized Partial Differential Equations, Springer Briefs Math. (2016) · Zbl 1329.65203 |
| [8] | Holmes, P.; Lumley, J. L.; Berkooz, G., Coherent Structures, Dynamical Systems and Symmetry, Cambridge Monogr. Mech. (1996), Cambridge University Press: Cambridge University Press Cambridge, UK · Zbl 0890.76001 |
| [9] | Little, G.; Reade, J. B., Eigenvalues of analytic kernels, SIAM J. Math. Anal., 15, 133-136 (1984) · Zbl 0536.45004 |
| [10] | Loève, M. M., Probability Theory (1988), Van Nostrand: Van Nostrand Princeton, NJ · Zbl 0108.14202 |
| [11] | Muller, M., On the POD Method. An Abstract Investigation with Applications to Reduced-Order Modeling and Suboptimal Control (2008), Georg-August Universität: Georg-August Universität Göttingen, PhD Thesis |
| [12] | Quarteroni, A.; Manzoni, A.; Negri, F., Reduced Basis Methods for Partial Differential Equations (2016), Springer · Zbl 1337.65113 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
