A quasi-optimal non-overlapping domain decomposition algorithm for the Helmholtz equation

This paper presents a new non-overlapping domain decomposition method for the Helmholtz equation, whose effective convergence is quasi-optimal. These improved properties result from a combination of an appropriate choice of transmission conditions and a suitable approximation of the Dirichlet to Neumann operator. A convergence theorem of the algorithm is established and numerical results validating the new approach are presented in both two and three dimensions.

[1]  Y. Lu.  Some Techniques for Computing Wave Propagation in Optical Waveguides , 2006 .

[2]  Patrick Joly,et al.  Domain Decomposition Method for Harmonic Wave Propagation : A General Presentation , 2000 .

[3]  Xavier Antoine,et al.  Numerical accuracy of a Padé‐type non‐reflecting boundary condition for the finite element solution of acoustic scattering problems at high‐frequency , 2005 .

[4]  G. G. Stokes.  "J." , 1890, The New Yale Book of Quotations.

[5]  Yassine Boubendir,et al.  Non-overlapping Domain Decomposition Method for a Nodal Finite Element Method , 2006, Numerische Mathematik.

[6]  M. Gunzburger,et al.  Boundary conditions for the numerical solution of elliptic equations in exterior regions , 1982 .

[7]  X. Antoine,et al.  Alternative integral equations for the iterative solution of acoustic scattering problems , 2005 .

[8]  Y. Lu,et al.  An improved surface radiation condition for high-frequency acoustic scattering problems , 2006 .

[9]  Gary H. Brooke,et al.  Rational square-root approximations for parabolic equation algorithms , 1997 .

[10]  Charbel Farhat,et al.  A two-level domain decomposition method for the iterative solution of high frequency exterior Helmholtz problems , 2000, Numerische Mathematik.

[11]  C. Farhat,et al.  Two-level domain decomposition methods with Lagrange multipliers for the fast iterative solution of acoustic scattering problems , 2000 .

[12]  E. Turkel,et al.  Absorbing PML boundary layers for wave-like equations , 1998 .

[13]  M. Guddati,et al.  Continued fraction absorbing boundary conditions for convex polygonal domains , 2006 .

[14]  Christophe Geuzaine,et al.  Gmsh: A 3‐D finite element mesh generator with built‐in pre‐ and post‐processing facilities , 2009 .

[15]  M. Czubak,et al.  PSEUDODIFFERENTIAL OPERATORS , 2020, Introduction to Partial Differential Equations.

[16]  Martin J. Gander,et al.  Optimized Schwarz Methods without Overlap for the Helmholtz Equation , 2002, SIAM J. Sci. Comput..

[17]  J. Remacle,et al.  Gmsh: A 3‐D finite element mesh generator with built‐in pre‐ and post‐processing facilities , 2009 .

[18]  Lonny L. Thompson,et al.  Parallel iterative solution for the Helmholtz equation with exact non-reflecting boundary conditions , 2006 .

[19]  Charbel Farhat,et al.  A domain decomposition method for discontinuous Galerkin discretizations of Helmholtz problems with plane waves and Lagrange multipliers , 2009 .

[20]  Alfredo Bermúdez,et al.  An Exact Bounded Perfectly Matched Layer for Time-Harmonic Scattering Problems , 2007, SIAM J. Sci. Comput..

[21]  Yassine Boubendir,et al.  An analysis of the BEM-FEM non-overlapping domain decomposition method for a scattering problem , 2007 .

[22]  X. Antoine,et al.  GENERALIZED COMBINED FIELD INTEGRAL EQUATIONS FOR THE ITERATIVE SOLUTION OF THE THREE-DIMENSIONAL HELMHOLTZ EQUATION , 2007 .

[23]  Charbel Farhat,et al.  A non Overlapping Domain Decomposition Method for the Exterior Helmholtz Problem , 1997 .

[24]  Peter Monk,et al.  The Perfectly Matched Layer in Curvilinear Coordinates , 1998, SIAM J. Sci. Comput..

[25]  Olof B. Widlund,et al.  29. Optimization of Interface Operator Based on Algebraic Approach , 2003 .

[26]  J. Volakis,et al.  Approximate boundary conditions in electromagnetics , 1995 .

[27]  Patrick Joly,et al.  Mathematical and Numerical Aspects of Wave Propagation Phenomena , 1991 .

[28]  A. Majda,et al.  Absorbing boundary conditions for the numerical simulation of waves , 1977 .

[29]  Xavier Antoine,et al.  A Construction of Beam Propagation Methods for Optical Waveguides , 2009 .

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

[31]  Yassine Boubendir,et al.  Coupling of a non‐overlapping domain decomposition method for a nodal finite element method with a boundary element method , 2008 .

[32]  Xavier Antoine,et al.  A performance study of plane wave finite element methods with a Padé-type artificial boundary condition in acoustic scattering , 2009, Adv. Eng. Softw..

[33]  Xavier Antoine,et al.  Bayliss-Turkel-like radiation conditions on surfaces of arbitrary shape , 1999 .

[34]  Christophe Geuzaine,et al.  A general environment for the treatment of discrete problems and its application to the finite element method , 1998 .

[35]  Ya Yan Lu,et al.  Improving the Beam Propagation Method for TM Polarization , 2002 .

[36]  I. Babuska,et al.  Dispersion analysis and error estimation of Galerkin finite element methods for the Helmholtz equation , 1995 .

[37]  Bruno Després.  Méthodes de décomposition de domaine pour la propagation d'ondes en régime harmonique. Le théorème de Borg pour l'équation de Hill vectorielle , 1991 .

[38]  Patrick Amestoy,et al.  A Fully Asynchronous Multifrontal Solver Using Distributed Dynamic Scheduling , 2001, SIAM J. Matrix Anal. Appl..

[39]  Yassine Boubendir,et al.  A FETI-like domain decomposition method for coupling finite elements and boundary elements in large-size problems of acoustic scattering , 2007 .

[40]  Frédéric Magoulès,et al.  Approximation of Optimal Interface Boundary Conditions for Two-Lagrange Multiplier FETI Method , 2005 .

[41]  A. Bayliss,et al.  Radiation boundary conditions for wave-like equations , 1980 .

[42]  Barry Smith,et al.  Domain Decomposition Methods for Partial Differential Equations , 1997 .

[43]  Bruno Stupfel,et al.  A hybrid finite element and integral equation domain decomposition method for the solution of the 3-D scattering problem , 2001 .

[44]  Christophe Geuzaine,et al.  Phase reduction models for improving the accuracy of the finite element solution of time-harmonic scattering problems I: General approach and low-order models , 2009, J. Comput. Phys..

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

[46]  I. Singer,et al.  A perfectly matched layer for the Helmholtz equation in a semi-infinite strip , 2004 .

[47]  F. Ihlenburg.  Finite Element Analysis of Acoustic Scattering , 1998 .