A BDDC algorithm with deluxe scaling for H(curl) in two dimensions with irregular subdomains

Abstract. A bound is obtained for the condition number of a BDDC algorithm for problems posed in H(curl) in two dimensions, where the subdomains are only assumed to be uniform in the sense of Peter Jones. For the primal variable space, a continuity constraint for the tangential average over each interior subdomain edge is imposed. For the averaging operator, a new technique named deluxe scaling is used. Our bound is independent of jumps in the coefficients across the interface between the subdomains and depends only on a few geometric parameters of the decomposition. Numerical results that verify the result are shown, including some with subdomains with fractal edges and others obtained by a mesh partitioner.

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

[2]  Andrea Toselli,et al.  Overlapping and Multilevel Schwarz Methods for Vector Valued Elliptic Problems in Three Dimensions , 2000 .

[3]  J. Nédélec.  Mixed finite elements in ℝ3 , 1980 .

[4]  Barbara I. Wohlmuth,et al.  An Iterative Substructuring Method for Raviart-Thomas Vector Fields in Three Dimensions , 2000, SIAM J. Numer. Anal..

[5]  Clark R. Dohrmann,et al.  An Iterative Substructuring Algorithm for Two-Dimensional Problems in H(curl) , 2012, SIAM J. Numer. Anal..

[6]  Susanne C. Brenner,et al.  BDDC and FETI-DP without matrices or vectors , 2007 .

[7]  Duk-Soon Oh,et al.  An Overlapping Schwarz Algorithm for Raviart-Thomas Vector Fields with Discontinuous Coefficients , 2013, SIAM J. Numer. Anal..

[8]  Olof B. Widlund,et al.  Some Recent Tools and a BDDC Algorithm for 3D Problems in H(curl) , 2013, Domain Decomposition Methods in Science and Engineering XX.

[9]  Peter W. Jones.  Quasiconformal mappings and extendability of functions in sobolev spaces , 1981 .

[10]  Wendell H. Fleming,et al.  Normal and Integral Currents , 1960 .

[11]  A. Quarteroni,et al.  Numerical Approximation of Partial Differential Equations , 2008 .

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

[13]  Olof B. Widlund,et al.  Isogeometric BDDC Preconditioners with Deluxe Scaling , 2014, SIAM J. Sci. Comput..

[14]  Olof B. Widlund,et al.  Accomodating Irregular Subdomains in Domain Decomposition Theory , 2009 .

[15]  Olof B. Widlund,et al.  BDDC Algorithms for Incompressible Stokes Equations , 2006, SIAM J. Numer. Anal..

[16]  CLARK R. DOHRMANN,et al.  A Preconditioner for Substructuring Based on Constrained Energy Minimization , 2003, SIAM J. Sci. Comput..

[17]  Dirk Pflüger,et al.  Lecture Notes in Computational Science and Engineering , 2010 .

[18]  Olof B. Widlund,et al.  FETI‐DP, BDDC, and block Cholesky methods , 2006 .

[19]  Juan Gabriel,et al.  A two-level overlapping Schwarz method for H(curl) in two dimensions with irregular subdomains , 2015 .

[20]  Olof B. Widlund,et al.  Domain Decomposition for Less Regular Subdomains: Overlapping Schwarz in Two Dimensions , 2008, SIAM J. Numer. Anal..

[21]  Jong Ho Lee,et al.  A Balancing Domain Decomposition by Constraints Deluxe Method for Reissner-Mindlin Plates with Falk-Tu Elements , 2015, SIAM J. Numer. Anal..

[22]  A. Bossavit.  Discretization of Electromagnetic Problems: The “Generalized Finite Differences” Approach , 2005 .

[23]  Jong Ho Lee,et al.  A BALANCING DOMAIN DECOMPOSITION BY CONSTRAINTS DELUXE METHOD FOR NUMERICALLY THIN REISSNER-MINDLIN PLATES APPROXIMATED WITH FALK-TU FINITE ELEMENTS TR2013-958 , 2013 .

[24]  Olof B. Widlund,et al.  An Alternative Coarse Space for Irregular Subdomains and an Overlapping Schwarz Algorithm for Scalar Elliptic Problems in the Plane , 2011, SIAM J. Numer. Anal..

[25]  Pekka Koskela,et al.  Sobolev-Poincaré implies John , 1995 .

[26]  Barbara I. Wohlmuth,et al.  An iterative substructuring method for Maxwell's equations in two dimensions , 2001, Math. Comput..

[27]  Jinchao Xu,et al.  Nodal Auxiliary Space Preconditioning in H(curl) and H(div) Spaces , 2007, SIAM J. Numer. Anal..

[28]  Andrea Toselli,et al.  Overlapping Schwarz methods for Maxwell's equations in three dimensions , 1997, Numerische Mathematik.

[29]  Olof B. Widlund,et al.  A BDDC Algorithm with Deluxe Scaling for Three‐Dimensional H(curl) Problems , 2016 .

[30]  Clark R. Dohrmann,et al.  Convergence of a balancing domain decomposition by constraints and energy minimization , 2002, Numer. Linear Algebra Appl..

[31]  Andrea Toselli,et al.  Robust and efficient FETI domain decomposition algorithms for edge element approximations , 2005 .

[32]  D. O’Leary,et al.  Capacitance Matrix Methods for the Helmholtz Equation on General Three-Dimensional Regions , 2015 .

[33]  Andrea Toselli,et al.  A FETI Domain Decomposition Method for Edge Element Approximations in Two Dimensions with Discontinuous Coefficients , 2001, SIAM J. Numer. Anal..

[34]  R. Hoppe,et al.  Residual based a posteriori error estimators for eddy current computation , 2000 .

[35]  B. Bojarski,et al.  Remarks on Sobolev imbedding inequalities , 1988 .

[36]  Yousef Saad,et al.  Iterative methods for sparse linear systems , 2003 .

[37]  J. Mandel,et al.  An algebraic theory for primal and dual substructuring methods by constraints , 2005 .

[38]  Duk-Soon Oh,et al.  A BDDC Algorithm for Raviart-Thomas Vector Fields. , 2013 .

[39]  Andrea Toselli,et al.  Dual-primal FETI algorithms for edge finite-element approximations in 3D , 2006 .

[40]  Vipin Kumar,et al.  A Fast and High Quality Multilevel Scheme for Partitioning Irregular Graphs , 1998, SIAM J. Sci. Comput..