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Robust operator estimates and the application to substructuringmethods for first-order systems

Published online by Cambridge University Press:  13 August 2014

Christian Wieners  and
Barbara Wohlmuth
Christian Wieners
Affiliation:
Institut für Angewandte und Numerische Mathematik, KIT, Karlsruhe, Germany. . christian.wieners@kit.edu
Barbara Wohlmuth
Affiliation:
Fakultät Mathematik M2, Technische Universität München, Garching, Germany. ; wohlmuth@ma.tum.de
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Abstract

We discuss a family of discontinuous Petrov–Galerkin (DPG) schemes for quite generalpartial differential operators. The starting point of our analysis is the DPG methodintroduced by [Demkowicz et al., SIAM J. Numer. Anal.49 (2011) 1788–1809; Zitelli et al., J.Comput. Phys. 230 (2011) 2406–2432]. This discretization resultsin a sparse positive definite linear algebraic system which can be obtained from a saddlepoint problem by an element-wise Schur complement reduction applied to the test space.Here, we show that the abstract framework of saddle point problems and domaindecomposition techniques provide stability and a priori estimates. Toobtain efficient numerical algorithms, we use a second Schur complement reduction appliedto the trial space. This restricts the degrees of freedom to the skeleton. We construct apreconditioner for the skeleton problem, and the efficiency of the discretization and thesolution method is demonstrated by numerical examples.

Keywords

First-order systemsPetrov–Galerkin methodssaddle point problems
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 48 , Issue 5 , September 2014 , pp. 1473 - 1494
Copyright
© EDP Sciences, SMAI 2014

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References

Adler, J.H., Brannick, J., Liu, C., Manteuffel, T. and Zikatanov, L., First-order system least squares and the energetic variational approach for two-phase flow. J. Comput. Phys. 230 (2011) 6647–6663. Google Scholar
Adler, J.H., Manteuffel, T.A., McCormick, S.F., Nolting, J.W., Ruge, J.W. and Tang, L., Efficiency based adaptive local refinement for first-order system least-squares formulations. SIAM J. Sci. Comput. 33 (2011) 1–24. Google Scholar
A. Barker, S. Brenner, E.-H. Park and L-Y. Sung, A one-level additive schwarz preconditioner for a discontinuous petrov-galerkin method. Preprint arXiv:1212.2645 (2012). To appear in the Proceeding of DD21.
Bochev, P.B. and Gunzburger, M.D., Finite element methods of least-squares type. SIAM Rev. 40 (1998) 789–837. Google Scholar
P.B. Bochev and M.D. Gunzburger, Least-Squares Finite Element Methods, vol. 166 of Appl. Math. Sci. Springer, New York (2009).
D. Braess, Finite Elements. Theory, fast solvers, and applications in solid mechaics. 3th ed. Cambridge University Press (2007).
Bramble, J.H., Lazarov, R.D. and Pasciak, J.E., A least-squares approach based on a discrete minus one inner product for first order systems. Math. Comput. 66 (1997) 935–955. Google Scholar
F. Brezzi and M. Fortin, Mixed and Hybrid Finite Element Methods. Springer (1991).
Bui-Thanh, T., Demkowicz, L. and Ghattas, O., A Unified Discontinuous Petrov−Galerkin Method and its Analysis for Friedrichs’ Systems. SIAM J. Numer. Anal. 51 (2013) 1933–1956. Google Scholar
Buffa, A. and Monk, P., Error estimates for the ultra weak variational formulation of the Helmholtz equation. Math. Model. Numer. Anal. 42 (2008) 925–940. Google Scholar
Cai, Z., Lazarov, R., Manteuffel, T.A. and McCormick, S.F., First-Order System Least Squares for Second-Order Partial Differential Equations: Part I. SIAM J. Numer. Anal. 31 (1994) 1785–1799. Google Scholar
Chan, J., Demkowicz, L. and Heuer, N., Robust DPG method for convection-dominated diffusion problems II: Natural inflow condition. Comput. Math. Appl. 67 (2014) 771–795. Google Scholar
Dahmen, W., Huang, C., Schwab, C. and Welper, G., Adaptive Petrov-Galerkin methods for first order transport equations. SIAM J. Numer. Anal. 50 (2012) 2420–2445. Google Scholar
W. Dahmen, C. Plesken and G. Welper, Double greedy algorithms: reduced basis methods for transport dominated problems (2013). Preprint arXiv:1302.5072.
Demkowicz, L. and Gopalakrishnan, J., Analysis of the DPG method for the Poisson equation. SIAM J. Numer. Anal. 49 (2011) 1788–1809. Google Scholar
Demkowicz, L., Gopalakrishnan, J., Muga, I. and Zitelli, J., Wavenumber explicit analysis for a DPG method for the multidimensional Helmholtz equation. Comput. Methods Appl. Mech. Engrg. 213 (2012) 126–138. Google Scholar
Demkowicz, L., Gopalakrishnan, J. and Niemi, A.H., A class of discontinuous Petrov–Galerkin methods. Part III: Adaptivity. Appl. Numer. Math. 62 (2012) 396–427. Google Scholar
Demkowicz, L. and Heuer, N., Robust DPG method for convection-dominated diffusion problems. SIAM J. Numer. Anal. 51 (2013) 2514–2537. Google Scholar
S. Esterhazy and J.M. Melenk, On stability of discretizations of the Helmholtz equation, in Numerical Analysis of Multiscale Problems, vol. 83 of Lect. Notes Comput. Sci. Engrg. Springer, Berlin (2012) 285–324.
J. Gopalakrishnan and W. Qiu, An analysis of the practical DPG method. Math. Comput. (2013).
Herrera, I., Trefftz method: A general theory. Numer. Methods Partial Differ. Eqs. 16 (2000) 561–580. Google Scholar
Heys, J.J., Lee, E., Manteuffel, T.A., Mccormick, S.F. and Ruge, J.W., Enhanced mass conservation in least-squares methods for Navier-Stokes equations. SIAM J. Sci. Comput. 31 (2009) 2303–2321. Google Scholar
Hiptmair, R., Moiola, A. and Perugia, I., Stability results for the time-harmonic Maxwell equations with impedance boundary conditions. Math. Models Methods Appl. Sci. 21 (2011) 2263–2287. Google Scholar
Hiptmair, R. and Xu, J., Nodal auxiliary space preconditioning in H(curl) and H(div) spaces. SIAM J. Numer. Anal. 45 (2007) 2483–2509. Google Scholar
B.N. Khoromskij and G. Wittum, Numerical solution of elliptic differential equations by reduction to the interface. Berlin, Springer (2004).
Krendl, W., Simoncini, V. and Zulehner, W., Stability Estimates and Structural Spectral Properties of Saddle Point Problems. Numer. Math. 124 (2013) 183–213. Google Scholar
Langer, U., Of, G., Steinbach, O. and Zulehner, W., Inexact data-sparse boundary element tearing and interconnecting methods. SIAM J. Sci. Comput. 29 (2007) 290–314. Google Scholar
J.M. Melenk, On generalized finite element methods. Ph.D. thesis, University of Maryland (1995).
A. Moiola, Trefftz-Discontinuous Galerkin Methods for Time-Harmonic Wave Problems. Ph.D. thesis, ETH Zürich (2011).
N. Roberts, T. Bui-Thanh and L. Demkowicz. The DPG method for the Stokes problem ICES Report (2012) 12–22.
Szyld, D.B., The many proofs of an identity on the norm of oblique projections. Numer. Algorithms 42 (2006) 309–323. Google Scholar
Wieners, C., A geometric data structure for parallel finite elements and the application to multigrid methods with block smoothing. Comput. Visual. Sci. 13 (2010) 161–175. Google Scholar
Xu, J. and Zikatanov, L., Some observations on Babuška and Brezzi theories. Numer. Math. 94 (2003) 195–202. Google Scholar
Zitelli, J., Muga, I., Demkowicz, L., Gopalakrishnan, J., Pardo, D. and Calo, V., A class of discontinuous Petrov−Galerkin methods. Part IV: Wave propagation. J. Comput. Phys. 230 (2011) 2406–2432. Google Scholar