A Calderón multiplicative preconditioner for the electromagnetic Poincaré-Steklov operator of a heterogeneous domain with scattering applications

In the context of hybrid formulations, the Poincare-Steklov operator acting on traces of solutions to the vector Helmholtz equation in a heterogeneous interior domain with a smooth boundary is regularized by a well-known boundary integral operator related to the homogeneous exterior domain. For the first time, this property allows us to simultaneously construct a Calderon multiplicative preconditioner for the discretized operator and for a 3-D hybrid finite/boundary element method formulation, applicable to electromagnetic scattering problems. Numerical examples demonstrate the effectiveness of this novel preconditioning scheme, even for heterogeneous domains with non-smooth boundaries.

[1]  John B. Shoven,et al.  I , Edinburgh Medical and Surgical Journal.

[2]  Benjamin S. Kirk,et al.  Library for Parallel Adaptive Mesh Refinement / Coarsening Simulations , 2006 .

[3]  Leslie Greengard,et al.  A fast algorithm for particle simulations , 1987 .

[4]  Weng Cho Chew,et al.  A Calderón Preconditioner for the Electric Field Integral Equation With Layered Medium Green's Function , 2014, IEEE Transactions on Antennas and Propagation.

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

[6]  J. Bladel,et al.  Electromagnetic Fields , 1985 .

[7]  D. Zutter,et al.  ON THE COMPLEX SYMMETRY OF THE POINCARÉ-STEKLOV OPERATOR , 2008 .

[8]  Ignace Bogaert,et al.  On a Well-Conditioned Electric Field Integral Operator for Multiply Connected Geometries , 2013, IEEE Transactions on Antennas and Propagation.

[9]  Francesco P. Andriulli,et al.  A Calderón Multiplicative Preconditioner for the PMCHWT integral equation , 2009, 2009 IEEE Antennas and Propagation Society International Symposium.

[10]  R. Hiptmair,et al.  Galerkin Boundary Element Methods for Electromagnetic Scattering , 2003 .

[11]  Philipp Birken,et al.  Numerical Linear Algebra , 2011, Encyclopedia of Parallel Computing.

[12]  H. Bağcı,et al.  Low-Frequency Scaling of the Standard and Mixed Magnetic Field and Müller Integral Equations , 2014, IEEE Transactions on Antennas and Propagation.

[13]  John L. Volakis,et al.  Frequency Domain Hybrid Finite Element Methods for Electromagnetics , 2006, Frequency Domain Hybrid Finite Element Methods for Electromagnetics.

[14]  Mario A. Storti,et al.  A preconditioner for the Schur complement matrix , 2006, Adv. Eng. Softw..

[15]  Roland W. Freund,et al.  A Transpose-Free Quasi-Minimal Residual Algorithm for Non-Hermitian Linear Systems , 1993, SIAM J. Sci. Comput..

[16]  Yassine Boubendir,et al.  Well-conditioned boundary integral equation formulations for the solution of high-frequency electromagnetic scattering problems , 2013, Comput. Math. Appl..

[17]  D. Wilton,et al.  Electromagnetic scattering by surfaces of arbitrary shape , 1980 .

[18]  Per-Gunnar Martinsson,et al.  A spectrally accurate direct solution technique for frequency-domain scattering problems with variable media , 2013, 1308.5998.

[19]  R. Kress,et al.  Inverse Acoustic and Electromagnetic Scattering Theory , 1992 .

[20]  Carretera de Valencia,et al.  The finite element method in electromagnetics , 2000 .

[21]  Jun Zhang,et al.  Sparse inverse preconditioning of multilevel fast multipole algorithm for hybrid Integral equations in electromagnetics , 2004 .

[22]  G. Mie.  Beiträge zur Optik trüber Medien, speziell kolloidaler Metallösungen , 1908 .

[23]  Ronald H. W. Hoppe,et al.  Finite element methods for Maxwell's equations , 2005, Math. Comput..

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

[25]  Daniël De Zutter,et al.  On the Complex Symmetry of the Poincar'E-Steklov Operator , 2008 .

[26]  E. Michielssen,et al.  A CalderÓn Multiplicative Preconditioner for the Combined Field Integral Equation , 2009, IEEE Transactions on Antennas and Propagation.

[27]  Daniël De Zutter,et al.  Accurate and Conforming Mixed Discretization of the MFIE , 2011, IEEE Antennas and Wireless Propagation Letters.

[28]  G. Burton.  Sobolev Spaces , 2013 .

[29]  X. Sheng,et al.  Hybrid h- and p-Type Multiplicative Schwarz (h-p-MUS) Preconditioned Algorithm of Higher-Order FE-BI-MLFMA for 3D Scattering , 2012, IEEE Transactions on Magnetics.

[30]  E. Michielssen,et al.  A Calderón-Preconditioned Single Source Combined Field Integral Equation for Analyzing Scattering From Homogeneous Penetrable Objects , 2011, IEEE Transactions on Antennas and Propagation.

[31]  Snorre H. Christiansen,et al.  A dual finite element complex on the barycentric refinement , 2005, Math. Comput..

[32]  P. Yla-Oijala,et al.  Calderon Preconditioned Surface Integral Equations for Composite Objects With Junctions , 2011, IEEE Transactions on Antennas and Propagation.

[33]  E. Michielssen,et al.  Nullspaces of MFIE and CalderÓn Preconditioned EFIE Operators Applied to Toroidal Surfaces , 2009, IEEE Transactions on Antennas and Propagation.

[34]  Giuseppe Vecchi,et al.  Solving the EFIE at Low Frequencies With a Conditioning That Grows Only Logarithmically With the Number of Unknowns , 2010, IEEE Transactions on Antennas and Propagation.

[35]  Alexander B. Yakovlev,et al.  Operator Theory for Electromagnetics: An Introduction , 2001 .

[36]  A Calderón preconditioner for the EFIE operator without barycentric refinement of the mesh , 2014, 2014 IEEE Antennas and Propagation Society International Symposium (APSURSI).

[37]  R. Harrington.  Boundary integral formulations for homogeneous material bodies , 1989 .

[38]  R. K. Luneburg,et al.  Mathematical Theory of Optics , 1966 .

[39]  J. Swinburne.  Electromagnetic Theory , 1894, Nature.

[40]  Roger F. Harrington,et al.  Field computation by moment methods , 1968 .

[41]  Xin-Qing Sheng,et al.  An Effective Domain-Decomposition-Based Preconditioner for the FE-BI-MLFMA Method for 3D Scattering Problems , 2014, IEEE Transactions on Antennas and Propagation.

[43]  Mario Alberto Storti,et al.  On the efficiency and quality of numerical solutions in CFD problems using the interface strip preconditioner for domain decomposition methods , 2006 .

[44]  Hendrik Rogier,et al.  Rigorous Analysis of Internal Resonances in 3-D Hybrid FE-BIE Formulations by Means of the Poincaré–Steklov Operator , 2013, IEEE Transactions on Microwave Theory and Techniques.

[45]  Jian-Ming Jin,et al.  EFIE Analysis of Low-Frequency Problems With Loop-Star Decomposition and Calderón Multiplicative Preconditioner , 2010, IEEE Transactions on Antennas and Propagation.

[46]  Ralf Hiptmair,et al.  Coupling of Finite Elements and Boundary Elements in Electromagnetic Scattering , 2003, SIAM J. Numer. Anal..

[47]  C. Schwab,et al.  Boundary Element Methods , 2010 .

[48]  T. Eibert,et al.  Calculation of unbounded field problems in free space by a 3D FEM/BEM-hybrid approach , 1996 .

[49]  Alfio Quarteroni,et al.  Theory and Application of Steklov-Poincaré Operators for Boundary-Value Problems , 1991 .

[50]  Zydrunas Gimbutas,et al.  A wideband fast multipole method for the Helmholtz equation in three dimensions , 2006, J. Comput. Phys..

[51]  Jianming Jin,et al.  Calderón Preconditioner: From EFIE and MFIE to N-Müller Equations , 2010, IEEE Transactions on Antennas and Propagation.

[52]  J. Nédélec.  Acoustic and Electromagnetic Equations : Integral Representations for Harmonic Problems , 2001 .

[53]  V. Rokhlin.  Rapid Solution of Integral Equations of Scattering Theory , 1990 .

[54]  Ting-Zhu Huang,et al.  Solution to 3-D electromagnetic problems discretized by a hybrid FEM/MOM method , 2013, Comput. Phys. Commun..

[56]  J. Z. Zhu,et al.  The finite element method , 1977 .

[57]  H. Rogier,et al.  The Poincaré–Steklov Operator in Hybrid Finite Element-Boundary Integral Equation Formulations , 2011, IEEE Antennas and Wireless Propagation Letters.

[58]  F. Alouges,et al.  A stable well-conditioned integral equation for electromagnetism scattering , 2007 .

[59]  A direct hierarchical multilevel preconditioner for the solution of finite element-boundary integral equations , 2014, The 8th European Conference on Antennas and Propagation (EuCAP 2014).

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

[61]  R.J. Adams,et al.  Physical and analytical properties of a stabilized electric field integral equation , 2004, IEEE Transactions on Antennas and Propagation.

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

[63]  E. Michielssen,et al.  A Multiplicative Calderon Preconditioner for the Electric Field Integral Equation , 2008, IEEE Transactions on Antennas and Propagation.

[64]  Jian-Ming Jin,et al.  A highly effective preconditioner for solving the finite element-boundary integral matrix equation of 3-D scattering , 2002 .

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

[66]  W. S. Venturini.  Boundary Integral Equations , 1983 .

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