Domain decomposition preconditioners for the discontinuous Petrov–Galerkin method

Li, Xiang; Xu, Xuejun

ESAIM: Mathematical Modelling and Numerical Analysis , Tome 51 (2017) no. 3, pp. 1021-1044.

In this paper, we design some efficient domain decomposition preconditioners for the discontinuous Petrov–Galerkin (DPG) method. Due to the special properties of the DPG method, the boundary condition becomes crucial in both of its application and analysis. We mainly focus on one of the boundary conditions: the Robin boundary condition, which actually appears in some useful model problems like the Helmholtz equation. We first design a two-level additive Schwarz preconditioner for the Poisson equation with a Robin boundary condition and give a rigorous condition number estimate for the preconditioned algebraic system. Moreover we also construct an additive Schwarz preconditioner for solving the Helmholtz equation. Numerical results show that the condition number of the preconditioned system is independent of wavenumber $\omega$ and mesh size $h$.

D. Arnold, R. Falk and R. Winther, Preconditioning in H(div) and applications. Math. Comp. 66 (1997) 957–984. | DOI | MR | Zbl

A.T. Barker, S.C. Brenner, E.-H. Park and L.-Y. Sung, A one-level additive Schwarz preconditioner for a discontinuous Petrov–Galerkin method, in Domain Decomposition Methods in Science and Engineering XXI. Springer (2014) 417–425. | MR

F. Brezzi, J. Douglas Jr. and L. Donatella Marini, Two families of mixed finite elements for second order elliptic problems. Numer. Math. 47 (1985) 217–235. | DOI | MR | Zbl

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) 1933–1958. | DOI | MR | Zbl

A. Byfut et al., FFW documentation. Humboldt University of Berlin, Germany (2007).

C. Carstensen, L. Demkowicz and J. Gopalakrishnan, A posteriori error control for DPG methods. SIAM J. Numer. Anal. 52 (2014) 1335–1353. | DOI | Zbl

J. Chan, L. Demkowicz, R. Moser and N. Roberts, A class of discontinuous Petrov–Galerkin methods. Part V: Solution of 1D Burgers and Navier–Stokes equations. ICES Report, 29 (2010).

J. Chan, L. Demkowicz and R. Moser, A DPG method for steady viscous compressible flow. Comput. Fluids 98 (2014) 69–90. | DOI | Zbl

M. Dauge, Elliptic boundary value problems on corner domains, Lect. Notes Math. Springer-Verlag (1988). | Zbl

R. Dautray and J.-L. Lions, Mathematical Analysis and Numerical Methods for Science and Technology. Vol. 3 of Spectral Theory and Applications. Springer Science & Business Media (1999). | Zbl

L. Demkowicz and J. Gopalakrishnan, A class of discontinuous Petrov–Galerkin methods. Part I: The transport equation. Comput. Methods Appl. Mech. Eng. 199 (2010) 1558–1572. | DOI | Zbl

L. Demkowicz and J. Gopalakrishnan, Analysis of the DPG method for the Poisson equation. SIAM J. Numer. Anal. 49 (2011) 1788–1809. | DOI | Zbl

L. Demkowicz and J. Gopalakrishnan, A primal DPG method without a first-order reformulation. Comput. Math. Appl. 66 (2013) 1058–1064. | DOI

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

L. Demkowicz, J. Gopalakrishnan, I. Muga and J. Zitelli, Wavenumber explicit analysis for a DPG method for the multidimensional Helmholtz equation, ICES Report 11-24, The University of Texas at Austin, 2011. Comput. Methods Appl. Mech. Engrg. 213-216 (2012) 126–138. | Zbl

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. | DOI | Zbl

M. Dryja and O.B. Widlund, Domain decomposition algorithms with small overlap. SIAM J. Sci. Comput. 15 (1994) 604–620. | DOI | Zbl

M. Fortin and F. Brezzi, Mixed and Hybrid Finite Element Methods. Springer-Verlag, New York (1991). | Zbl

J. Gopalakrishnan and W. Qiu, An analysis of the practical DPG method. Math. Comput. 83 (2014) 537–552. | DOI | Zbl

J. Gopalakrishnan and J. Schöberl, Degree and wavenumber [in] dependence of Schwarz preconditioner for the DPG method, in Spectral and High Order Methods for Partial Differential Equations ICOSAHOM 2014. Springer (2015) 257–265.

R. Hiptmair, Multigrid method for Maxwell’s equations. SIAM J. Numer. Anal. 36 (1998) 204–225. | DOI | Zbl

R. Hiptmair and R.H.W. Hoppe, Multilevel methods for mixed finite elements in three dimensions. Numer. Math. 82 (1999) 253–279. | DOI | Zbl

R. Hiptmair and A. Toselli, Overlapping and multilevel Schwarz methods for vector valued elliptic problems in three dimensions, in Parallel Solution of Partial Differential Equations. Springer (2000) 181–208. | Zbl

J.-C. Nédélec, Mixed finite elements in R3. Numer. Math. 35 (1980) 315–341. | DOI | MR | Zbl

P.-A. Raviart and J.-M. Thomas, A mixed finite element method for 2-nd order elliptic problems, in Mathematical Aspects of Finite Element Methods. Springer (1977) 292–315. | Zbl

S. Reitzinger and J. Schöberl, An algebraic multigrid method for finite element discretizations with edge elements. Numer. Linear Algebra 9 (2002) 223–238. | DOI | Zbl

N.V. Roberts, T. Bui-Thanh and L. Demkowicz, The DPG method for the Stokes problem. Comput. Math. Appl. 67 (2014) 966–995. | DOI | Zbl

N.V. Roberts, L. Demkowicz and R. Moser, A discontinuous Petrov–Galerkin methodology for adaptive solutions to the incompressible Navier–Stokes equations. J. Comput. Phys. 301 (2015) 456–483. | DOI | Zbl

L. Ridgway Scott and S. Zhang, Finite element interpolation of nonsmooth functions satisfying boundary conditions. Math. Comput. 54 (1990) 483–493. | DOI | Zbl

A. Toselli and O. Widlund, Domain Decomposition Methods: Algorithms and Theory, vol. 3. Springer (2005). | Zbl