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

Wieners, Christian; Wohlmuth, Barbara

ESAIM: Mathematical Modelling and Numerical Analysis , Tome 48 (2014) no. 5, pp. 1473-1494.

We discuss a family of discontinuous Petrov-Galerkin (DPG) schemes for quite general partial differential operators. The starting point of our analysis is the DPG method introduced by [Demkowicz et al., SIAM J. Numer. Anal. 49 (2011) 1788-1809; Zitelli et al., J. Comput. Phys. 230 (2011) 2406-2432]. This discretization results in a sparse positive definite linear algebraic system which can be obtained from a saddle point 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 domain decomposition techniques provide stability and a priori estimates. To obtain efficient numerical algorithms, we use a second Schur complement reduction applied to the trial space. This restricts the degrees of freedom to the skeleton. We construct a preconditioner for the skeleton problem, and the efficiency of the discretization and the solution method is demonstrated by numerical examples.

Classification : 65N30
Mots clés : first-order systems, Petrov-Galerkin methods, saddle point problems

[1] J.H. Adler, J. Brannick, C. Liu, T. Manteuffel and L. Zikatanov, First-order system least squares and the energetic variational approach for two-phase flow. J. Comput. Phys. 230 (2011) 6647-6663. | MR

[2] J.H. Adler, T.A. Manteuffel, S.F. Mccormick, J.W. Nolting, J.W. Ruge and L. Tang, Efficiency based adaptive local refinement for first-order system least-squares formulations. SIAM J. Sci. Comput. 33 (2011) 1-24. | MR

[3] 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.

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

[5] P.B. Bochev and M.D. Gunzburger, Least-Squares Finite Element Methods, vol. 166 of Appl. Math. Sci. Springer, New York (2009). | MR | Zbl

[6] D. Braess, Finite Elements. Theory, fast solvers, and applications in solid mechaics. 3th ed. Cambridge University Press (2007). | Zbl

[7] J.H. Bramble, R.D. Lazarov and J.E. Pasciak, A least-squares approach based on a discrete minus one inner product for first order systems. Math. Comput. 66 (1997) 935-955. | MR | Zbl

[8] F. Brezzi and M. Fortin, Mixed and Hybrid Finite Element Methods. Springer (1991). | MR | Zbl

[9] T. Bui-Thanh, L. Demkowicz and O. Ghattas, A Unified Discontinuous Petrov−Galerkin Method and its Analysis for Friedrichs' Systems. SIAM J. Numer. Anal. 51 (2013) 1933-1956. | MR | Zbl

[10] A. Buffa and P. Monk, Error estimates for the ultra weak variational formulation of the Helmholtz equation. Math. Model. Numer. Anal. 42 (2008) 925-940. | Numdam | MR | Zbl

[11] Z. Cai, R. Lazarov, T.A. Manteuffel and S.F. Mccormick, First-Order System Least Squares for Second-Order Partial Differential Equations: Part I. SIAM J. Numer. Anal. 31 (1994) 1785-1799. | MR | Zbl

[12] J. Chan, L. Demkowicz and N. Heuer, Robust DPG method for convection-dominated diffusion problems II: Natural inflow condition. Comput. Math. Appl. 67 (2014) 771-795. | MR | Zbl

[13] W. Dahmen, C. Huang, C. Schwab and G. Welper, Adaptive Petrov-Galerkin methods for first order transport equations. SIAM J. Numer. Anal. 50 (2012) 2420-2445. | MR | Zbl

[14] W. Dahmen, C. Plesken and G. Welper, Double greedy algorithms: reduced basis methods for transport dominated problems (2013). Preprint arXiv:1302.5072. | Numdam | MR

[15] L. Demkowicz and J. Gopalakrishnan, Analysis of the DPG method for the Poisson equation. SIAM J. Numer. Anal. 49 (2011) 1788-1809. | MR | Zbl

[16] L. Demkowicz, J. Gopalakrishnan, I. Muga and J. Zitelli, Wavenumber explicit analysis for a DPG method for the multidimensional Helmholtz equation. Comput. Methods Appl. Mech. Engrg. 213 (2012) 126-138. | MR | Zbl

[17] L. Demkowicz, J. Gopalakrishnan and A.H. Niemi, A class of discontinuous Petrov-Galerkin methods. Part III: Adaptivity. Appl. Numer. Math. 62 (2012) 396-427. | MR

[18] L. Demkowicz and N. Heuer, Robust DPG method for convection-dominated diffusion problems. SIAM J. Numer. Anal. 51 (2013) 2514-2537. | MR | Zbl

[19] 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. | MR | Zbl

[20] J. Gopalakrishnan and W. Qiu, An analysis of the practical DPG method. Math. Comput. (2013). | MR | Zbl

[21] I. Herrera, Trefftz method: A general theory. Numer. Methods Partial Differ. Eqs. 16 (2000) 561-580. | MR | Zbl

[22] J.J. Heys, E. Lee, T.A. Manteuffel, S.F. Mccormick and J.W. Ruge, Enhanced mass conservation in least-squares methods for Navier-Stokes equations. SIAM J. Sci. Comput. 31 (2009) 2303-2321. | MR | Zbl

[23] R. Hiptmair, A. Moiola and I. Perugia, Stability results for the time-harmonic Maxwell equations with impedance boundary conditions. Math. Models Methods Appl. Sci. 21 (2011) 2263-2287. | MR

[24] R. Hiptmair and J. Xu, Nodal auxiliary space preconditioning in H(curl) and H(div) spaces. SIAM J. Numer. Anal. 45 (2007) 2483-2509. | MR | Zbl

[25] B.N. Khoromskij and G. Wittum, Numerical solution of elliptic differential equations by reduction to the interface. Berlin, Springer (2004). | MR | Zbl

[26] W. Krendl, V. Simoncini and W. Zulehner, Stability Estimates and Structural Spectral Properties of Saddle Point Problems. Numer. Math. 124 (2013) 183-213. | MR | Zbl

[27] U. Langer, G. Of, O. Steinbach and W. Zulehner, Inexact data-sparse boundary element tearing and interconnecting methods. SIAM J. Sci. Comput. 29 (2007) 290-314. | MR | Zbl

[28] J.M. Melenk, On generalized finite element methods. Ph.D. thesis, University of Maryland (1995). | MR

[29] A. Moiola, Trefftz-Discontinuous Galerkin Methods for Time-Harmonic Wave Problems. Ph.D. thesis, ETH Zürich (2011).

[30] N. Roberts, T. Bui-Thanh and L. Demkowicz. The DPG method for the Stokes problem ICES Report (2012) 12-22.

[31] D.B. Szyld, The many proofs of an identity on the norm of oblique projections. Numer. Algorithms 42 (2006) 309-323. | MR | Zbl

[32] C. Wieners, A geometric data structure for parallel finite elements and the application to multigrid methods with block smoothing. Comput. Visual. Sci. 13 (2010) 161-175. | MR | Zbl

[33] J. Xu and L. Zikatanov, Some observations on Babuška and Brezzi theories. Numer. Math. 94 (2003) 195-202. | MR | Zbl

[34] J. Zitelli, I. Muga, L. Demkowicz, J. Gopalakrishnan, D. Pardo and V. Calo, A class of discontinuous Petrov−Galerkin methods. Part IV: Wave propagation. J. Comput. Phys. 230 (2011) 2406-2432. | MR