Abstract
In this paper, we propose a certified reduced basis (RB) method for quasilinear parabolic problems with strongly monotone spatial differential operator. We provide a residual-based a posteriori error estimate for a space-time formulation and the corresponding efficiently computable bound for the certification of the method. We introduce a Petrov-Galerkin finite element discretization of the continuous space-time problem and use it as our reference in a posteriori error control. The Petrov-Galerkin discretization is further approximated by the Crank-Nicolson time-marching problem. It allows to use a POD-Greedy approach to construct the reduced-basis spaces of small dimensions and to apply the Empirical Interpolation Method (EIM) to guarantee the efficient offline-online computational procedure. In our approach, we compute the reduced basis solution in a time-marching framework while the RB approximation error in a space-time norm is controlled by our computable bound. Therefore, we combine a POD-Greedy approximation with a space-time Galerkin method.
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Alla, A., Hinze, M., Kolvenbach, P., et al.: A certified model reduction approach for robust parameter optimization with PDE constraints. Adv. Comput. Math. 45, 1221–1250 (2019)
Bachinger, F., Langer, U., Schöberl, J.: Numerical analysis of nonlinear multiharmonic eddy current problems. Numer. Math. 100(4), 593–616 (2005)
Barrault, M., Maday, Y., Nguyen, N.C., Patera, A.T.: An empirical interpolation method: application to efficient reduced-basis discretization of partial differential equations. C.R. Acad. Sci. Paris Ser.I. 339(9), 667–672 (2004)
Glas, S., Mayerhofer, A., Urban, K.: Two ways to treat time in reduced basis methods. Model Reduct. Parametr. syst. 17, 1–16 (2017)
Grepl, M.A.: Certified reduced basis methods for nonaffine linear time-varying and nonlinear parabolic partial differential equations. Math. Models Methods Appl. Sci. 22(03), 1150015 (2012)
Grepl, M.A., Maday, Y., Nguyen, N.C., Patera, A.T.: Efficient reduced-basis treatment of nonaffine and nonlinear partial differential equations. ESAIM: Math. Model. and Numer. Anal. 41(3), 575–605 (2007)
Grepl, M.A., Patera, A.T.: A posteriori error bounds for reduced-basis approximations of parametrized parabolic partial differential equations. ESAIM: Math. Model. and Numer. Anal. 39(1), 157–181 (2005)
Haasdonk, B., Ohlberger, M.: Reduced basis method for finite volume approximations of parametrized linear evolution equations. ESAIM: Math. Model. and Numeric. Anal. 42(2), 277–302 (2008)
Haasdonk, B.: Reduced basis methods for parametrized PDEs—a tutorial introduction for stationary and snstationary problems. In: Benner, P., Cohen, A., Ohlberger, M., Willcox, K (eds.) Chapter in model reduction and approximation: theory and algorithms, pp. 65–136. SIAM, Philadelphia (2017)
Heise, B.: Analysis of a fully discrete finite element method for a nonlinear magnetic field problem. SIAM J. Num. Anal. 31(3), 745–759 (1994)
Hinze, M., Korolev, D.: Reduced basis methods for quasilinear elliptic PDEs with applications to permanent magnet synchronous motors, Technical report: https://arxiv.org/pdf/2002.04288.pdf, (2020)
Ion, I.G., Bontinck, Z., Loukrezis, D., et al.: Robust shape optimization of electric devices based on deterministic optimization methods and finite-element analysis with affine parametrization and design elements. Electr. Eng. 100, 2635–2647 (2018)
Kerler-Back, J., Stykel, T.: Model reduction for linear and nonlinear magneto-quasistatic equations. Int. J. Numer. Methods Eng 111(13), 1274–1299 (2017)
Maday, Y., Nguyen, N. C., Patera, A. T., Pau, G. S. H.: A general multipurpose interpolation procedure: the magic points. Commun. Pure Appl. Anal. 8(1), 383–404 (2009)
Quarteroni, A., Manzoni, A., Negri, F.: Reduced basis methods for partial differential equations: an Introduction, vol. 92. Springer International Publishing, Switzerland (2016)
Rozza, G., Huynh, D.B.P., Patera, A.T.: Reduced basis approximation and a posteriori error estimation for affinely parametrized elliptic coercive partial differential equations. Arch. Comput. Methods Eng. 15(3), 229–275 (2008)
Salon, S.J.: Finite element analysis of electrical machines. Kluwer Academic Publishers, Boston (1995)
Schöps, S.: private communication (2020)
Steih, K., Urban, K.: Space-time reduced basis methods for time-periodic partial differential equations. IFAC Proc. Volumes 45(2), 710–715 (2012)
Urban, K., Patera, A.T.: An improved error bound for reduced basis approimation of linear parabolic problems. Math. Comp. 83(288), 1599–1615 (2014)
Volkwein, S.: Proper orthogonal decomposition: theory and reduced-order modelling, Lecture Notes, University of Konstanz (2013)
Yano, M., Patera, A.T., Urban, K.: A space-time hp-interpolation-based certified reduced basis method for Burgers’ equation. Math. Models Methods Appl. Sci. 24(09), 1903–1935 (2014)
Yano, M.: A space-time Petrov–Galerkin certified reduced basis method: application to the Boussinesq equations. SIAM J. Sci. Comput. 36(1), A232–A266 (2014)
Zeidler, E.: Nonlinear functional analysis and its applications i/a: linear monotone operators. Springer science + business media new york (1990)
Zeidler, E.: Nonlinear functional analysis and its applications II/b: nonlinear monotone operators. Springer science + business media new york (1990)
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Open Access funding enabled and organized by Projekt DEAL. Both authors acknowledge the support of the collaborative research project PASIROM funded by the German Federal Ministry of Education and Research (BMBF) under grant no. 05M2018.
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Communicated by: Anthony Patera
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Hinze, M., Korolev, D. A space-time certified reduced basis method for quasilinear parabolic partial differential equations. Adv Comput Math 47, 36 (2021). https://doi.org/10.1007/s10444-021-09860-z
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DOI: https://doi.org/10.1007/s10444-021-09860-z
Keywords
- Parametrized parabolic equations
- Certified reduced basis
- Space-time Petrov-Galerkin
- A posteriori error estimate
- Empirical interpolation method