A FETI-DP Preconditioner for a Composite Finite Element and Discontinuous Galerkin Method

In this paper a Nitsche-type discretization based on a discontinuous Galerkin (DG) method for an elliptic two-dimensional problem with discontinuous coefficients is considered. The problem is posed on a polygonal region $\Omega$ which is a union of $N$ disjoint polygonal subdomains $\Omega_i$ of diameter $O(H_i)$. The discontinuities of the coefficients, possibly very large, are assumed to occur only across the subdomain interfaces $\partial \Omega_i$. Inside each subdomain, a conforming finite element space on a quasi-uniform triangulation with mesh size $O(h_i)$ is introduced. To handle the nonmatching meshes across the subdomain interfaces, a DG discretization is applied only on the interfaces. For solving the resulting discrete system, a FETI--DP-type method is proposed and analyzed. It is established that the condition number of the preconditioned linear system is estimated by $C(1 + \max_i \log H_i/h_i)^2$ with a constant $C$ independent of $h_i$, $h_i/h_j$, $H_i$ and the jumps of coefficients. The ...

[1]  Xiaobing Feng,et al.  Two-Level Additive Schwarz Methods for a Discontinuous Galerkin Approximation of Second Order Elliptic Problems , 2001, SIAM J. Numer. Anal..

[2]  J. Galvis,et al.  NEUMANN-NEUMANN METHODS FOR A DG DISCRETIZATION OF ELLIPTIC PROBLEMS WITH DISCONTINUOUS COEFFICIENTS ON GEOMETRICALLY NONCONFORMING SUBSTRUCTURES , 2009 .

[3]  C. Bernardi,et al.  A New Nonconforming Approach to Domain Decomposition : The Mortar Element Method , 1994 .

[4]  C. Farhat,et al.  A scalable dual-primal domain decomposition method , 2000, Numer. Linear Algebra Appl..

[5]  Rolf Stenberg,et al.  MORTARING BY A METHOD OF J.A. NITSCHE , 1998 .

[6]  Paola F. Antonietti,et al.  Schwarz domain decomposition preconditioners for discontinuous Galerkin approximations of elliptic problems: non-overlapping case , 2007 .

[7]  Blanca Ayuso de Dios,et al.  Uniformly Convergent Iterative Methods for Discontinuous Galerkin Discretizations , 2009, J. Sci. Comput..

[8]  B. Heinrich,et al.  Nitsche Type Mortaring for some Elliptic Problem with Corner Singularities , 2002, Computing.

[9]  Douglas N. Arnold,et al.  Unified Analysis of Discontinuous Galerkin Methods for Elliptic Problems , 2001, SIAM J. Numer. Anal..

[10]  Ludmil T. Zikatanov,et al.  Two‐level preconditioning of discontinuous Galerkin approximations of second‐order elliptic equations , 2006, Numer. Linear Algebra Appl..

[11]  Satyendra K. Tomar,et al.  Multilevel Preconditioning of Two-dimensional Elliptic Problems Discretized by a Class of Discontinuous Galerkin Methods , 2008, SIAM J. Sci. Comput..

[12]  Guido Kanschat,et al.  Preconditioning Methods for Local Discontinuous Galerkin Discretizations , 2003, SIAM J. Sci. Comput..

[13]  Béatrice Rivière,et al.  Discontinuous Galerkin methods for solving elliptic and parabolic equations - theory and implementation , 2008, Frontiers in applied mathematics.

[14]  Stanimire Tomov,et al.  Interior Penalty Discontinuous Approximations of Elliptic Problems , 2001 .

[15]  Shun Zhang,et al.  Discontinuous Galerkin Finite Element Methods for Interface Problems: A Priori and A Posteriori Error Estimations , 2011, SIAM J. Numer. Anal..

[16]  Svetozar Margenov,et al.  CBS constants for multilevel splitting of graph-Laplacian and application to preconditioning of discontinuous Galerkin systems , 2007, J. Complex..

[17]  Paola F. Antonietti,et al.  Multiplicative Schwarz Methods for Discontinuous Galerkin Approximations ofElliptic Problems , 2007 .

[18]  L. Zikatanov,et al.  A Simple Uniformly Convergent Iterative Method for the Non-symmetric Incomplete Interior Penalty Discontinuous Galerkin Discretization , 2011 .

[19]  Susanne C. Brenner,et al.  Two-level additive Schwarz preconditioners for C0 interior penalty methods , 2005, Numerische Mathematik.

[20]  Olof B. Widlund,et al.  An Analysis of a FETI-DP Algorithm on Irregular Subdomains in the Plane , 2008, SIAM J. Numer. Anal..

[21]  Andrea Toselli,et al.  Domain decomposition methods : algorithms and theory , 2005 .

[22]  Jan Mandel,et al.  On the convergence of a dual-primal substructuring method , 2000, Numerische Mathematik.

[23]  M. Sarkis.  Nonstandard coarse spaces and Schwarz methods for elliptic problems with discontinuous coefficients using non-conforming elements , 1997 .

[24]  Juan Galvis,et al.  BDDC methods for discontinuous Galerkin discretization of elliptic problems , 2007, J. Complex..

[25]  Guido Kanschat,et al.  A multilevel discontinuous Galerkin method , 2003, Numerische Mathematik.

[26]  Satyendra K. Tomar,et al.  A multilevel method for discontinuous Galerkin approximation of three‐dimensional anisotropic elliptic problems , 2008, Numer. Linear Algebra Appl..

[27]  Olof B. Widlund,et al.  DUAL-PRIMAL FETI METHODS FOR THREE-DIMENSIONAL ELLIPTIC PROBLEMS WITH HETEROGENEOUS COEFFICIENTS , 2022 .

[28]  D. Rixen,et al.  FETI‐DP: a dual–primal unified FETI method—part I: A faster alternative to the two‐level FETI method , 2001 .

[29]  D. Arnold.  An Interior Penalty Finite Element Method with Discontinuous Elements , 1982 .

[30]  M. Dryja.  On Discontinuous Galerkin Methods for Elliptic Problems with Discontinuous Coefficients , 2003 .

[31]  Charbel Farhat,et al.  A scalable dual-primal domain decomposition method , 2000, Numerical Linear Algebra with Applications.

[32]  Juan Galvis,et al.  Neumann‐Neumann methods for a DG discretization on geometrically nonconforming substructures , 2012 .

[33]  Xiaobing Feng,et al.  Analysis of Two-Level Overlapping Additive Schwarz Preconditioners for a Discontinuous Galerkin Method , .

[34]  Andrea Toselli,et al.  An overlapping domain decomposition preconditioner for a class of discontinuous Galerkin approximations of advection-diffusion problems , 2000, Math. Comput..

[35]  J. Hesthaven,et al.  Nodal Discontinuous Galerkin Methods: Algorithms, Analysis, and Applications , 2007 .

[36]  O. Widlund,et al.  Multilevel Schwarz methods for elliptic problems with discontinuous coefficients in three dimensions , 1994 .