A minimum-residual mixed reduced basis method: Exact residual certification and simultaneous finite-element reduced-basis refinement

Yano, Masayuki

ESAIM: Mathematical Modelling and Numerical Analysis , Tome 50 (2016) no. 1, pp. 163-185.

We present a reduced basis method for parametrized partial differential equations certified by a dual-norm bound of the residual computed not in the typical finite-element “truth” space but rather in an infinite-dimensional function space. The bound builds on a finite element method and an associated reduced-basis approximation derived from a minimum-residual mixed formulation. The offline stage combines a spatial mesh adaptation for finite elements and a greedy parameter sampling strategy for reduced bases to yield a reliable online system in an efficient manner; the online stage provides the solution and the associated dual-norm bound of the residual for any parameter value in complexity independent of the finite element resolution. We assess the effectiveness of the approach for a parametrized reaction-diffusion equation and a parametrized advection-diffusion equation with a corner singularity; not only does the residual bound provide reliable certificates for the solutions, the associated mesh adaptivity significantly reduces the offline computational cost for the reduced-basis generation and the greedy parameter sampling ensures quasi-optimal online complexity.

M. Ainsworth and J.T. Oden, A posteriori error estimation in finite element analysis. Comput. Methods Appl. Mech. Engrg. 142 (1997) 1–88. | DOI | MR | Zbl

M. Barrault, Y. Maday, N.C. Nguyen and A.T. Patera, An “empirical interpolation” method: application to efficient reduced-basis discretization of partial differential equations. C. R. Acad. Sci. Paris, Ser. I. 339 (2004) 667–672. | DOI | MR | Zbl

P. Binev, A. Cohen, W. Dahmen, R. Devore, G. Petrova and P. Wojtaszczyk, Convergence rates for greedy algorithms in reduced basis methods. SIAM J. Math. Anal. 43 (2011) 1457–1472. | DOI | MR | Zbl

P.B. Bochev and M.D. Gunzburger, Finite element methods of least-squares type. SIAM Rev. 40 (1998) 789–837. | DOI | MR | Zbl

S. Chaturantabut and D.C. Sorensen, Nonlinear model reduction via Discrete Empirical Interpolation. SIAM J. Sci. Comput. 32 (2010) 2737–2764. | DOI | MR | Zbl

W. Dahmen, C. Plesken and G. Welper, Double greedy algorithms: reduced basis methods for transport dominated problems. Math. Model. Numer. Anal. 48 (2013) 623–663. | DOI | Numdam | MR | Zbl

M. Drohmann, B. Haasdonk and M. Ohlberger, Reduced basis approximation for nonlinear parametrized evolution equations based on empirical operator interpolation. SIAM J. Sci. Comput. 34 (2012) 934–969. | DOI | MR | Zbl

J.L. Eftang, A.T. Patera and E.M. Rønquist, An “$hp$” certified reduced basis method for parametrized elliptic partial differential equations. SIAM J. Sci. Comput. 32 (2010) 3170–3200. | DOI | MR | Zbl

P. Ladevèze and D. Leguillion, Error estimate procedure in the finite element method and applications. SIAM J. Numer. Anal. 20 (1983) 485–509. | DOI | MR | Zbl

P. Ladevèze and L. Chamoin, On the verification of model reduction methods based on the proper generalized decomposition. Comput. Methods Appl. Mech. Engrg. 200 (2011) 2032–2047. | DOI | MR | Zbl

Y. Maday and O. Mula, A Generalized Empirical Interpolation Method: Application of Reduced Basis Techniques to Data Assimilation. In Anal. Numer. Partial Differ. Equations, edited by F. Brezzi, P.C. Franzone, U. Gianazza and G. Gilardi. Springer-Verlag (2013) 221–235. | MR | Zbl

W.F. Mitchell, Adaptive refinement for arbitrary finite-element spaces with hierarchical bases. J. Comput. Appl. Math. 36 (1991) 65–78. | DOI | MR | Zbl

P. Morin, R.H. Nochetto and K.G. Siebert, Convergence of adaptive finite element methods. SIAM Rev. 44 (2002) 631–658. | DOI | MR | Zbl

A.I. Pehlivanov, G.F. Carey and R.D. Lazarov, Least-squares mixed finite elements for second-order elliptic problems. SIAM J. Numer. Anal. 31 (1994) 1368–1377. | DOI | MR | Zbl

A.I. Pehlivanov, G.F. Carey and P.S. Vassilevski, Least-squares mixed finite element methods for non-selfadjoint elliptic problems: I. Error estimates. Numer. Math. 72 (1996) 501–522. | DOI | MR | Zbl

P.A. Raviart and J.M. Thomas, A Mixed Finite Element Method for 2nd Order Elliptic Problems. In Vol. 606 of Lect. Notes Math. Springer-Verlag (1977) 292–315. | MR | Zbl

G. Rozza and K. Veroy, On the stability of the reduced basis method for stokes equations in parametrized domains. Comput. Methods Appl. Mech. Engrg. 196 (2007) 1244–1260. | DOI | MR | Zbl

G. Rozza, D.B.P. Huynh and 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. Methods Eng. 15 (2008) 229–275. | DOI | MR | Zbl

A.M. Sauer-Budge, J. Bonet, A. Huerta and J. Peraire, Computing bounds for linear functionals of exact weak solutions to Poisson’s equation. SIAM J. Numer. Anal. 42 (2004) 1610–1630. | DOI | MR | Zbl

C. Schwab, $p$- and $hp$-Finite Element Methods. Oxford Science Publications, Great Clarendon Street, Oxford, UK (1998). | MR | Zbl

K. Steih and K. Urban, Reduced basis methods based upon adaptive snapshot computations. Preprint arxiv: 1407.1708v1 (2014).

T. Tonn, K. Urban and S. Volkwein, Comparison of the reduced-basis and POD a posteriori error estimators for an elliptic linear-quadratic optimal control problem. Math. Comput. Model. Dyn. 17 (2011) 355–369. | DOI | MR | Zbl

M. Yano, A reduced basis method with exact-solution certificates for steady symmetric coercive equations. Comput. Methods Appl. Mech. Engrg. 287 (2015) 290–309. | DOI | MR | Zbl