Data-sparse Boundary and Finite Element Domain Decomposition Methods in Electromagnetics

Funding agency	Austrian Science Fund (FWF)
Project number	P 19255
Principal investigator	Urlich Langer
International Research Partner and Co-Investigator	Olaf Steinbach
Members and basic staff	Sven Beuchler Dylan Copeland Sarah Engleder Peter Gruber Michael Kolmbauer Günther Of Clemens Pechstein Markus Windisch Monika Wolfmayr (née Kowalska) Sabine Zaglmayr Walter Zulehner
Duration	2007-05-01 – 2012-04-30

This website documents the work in the project Data-sparse Boundary and Finite Element Domain Decomposition Methods in Electromagnetics, which is supported by the Austrian Science Fund (FWF) under grant P19255-N18. This is a joint project of the Institute of Computational Mathematics at the Johannes Kepler University in Linz and the Institute of Computational Mathematics at the University of Technology in Graz.

Description

Domain Decomposition (DD) methods are nowadays not only used for constructing parallel solvers for Partial Differential Equations (PDE), but also for coupling different physical fields and different discretization techniques. Since Finite Element Methods (FEM) and Boundary Element Methods (BEM) exhibit certain complementary properties, it is sometimes very useful to couple these discretization techniques and thus benefit from the advantages of both worlds. This concerns not only the treatment of unbounded domains (BEM), but also the right handling of singularities (BEM), moving parts (BEM), air regions in electromagnetics (BEM), source terms (FEM), non-linearities (FEM), etc. Therefore, it is not astonishing that the coupling of FEM and BEM within a DD framework is successfully used in many practical applications. Among the DD methods, the so-called Finite Element Tearing and Interconnecting (FETI) methods are probably the most successful, at least for large-scale parallel computations. Recently, the proposers have introduced data-sparse Boundary Element Tearing and Interconnecting (BETI) methods as boundary element counterparts of the well-established FETI methods, as well as coupled BETI-FETI methods for some model problems such as the potential equation and the linear elasticity system. The advantage of these tearing and interconnecting methods is not only the nearly optimal asymptotical behavior of the iteration numbers with respect to the discretization parameter h and the subdomain scaling parameter H, but also the robustness with respect to large coefficient jumps and the excellent scalability on massively parallel computers.

In this project, we propose to construct and analyze new DD solvers, including DD solvers based on the tearing and interconnecting technique, for large-scale finite element (FE), boundary element (BE), and coupled FE-BE DD equations derived from linear and non-linear magnetostatic problems as well as from linear and non-linear eddy current problems in both the time and frequency domains. The numerical treatment of non-linear eddy current problems in the frequency domain is not at all straightforward. A multiharmonic approach that is based on Fourier series is one possible technique to treat such problems. The construction of fast solvers, in particular, efficient DD solvers for the resulting large-scale system of non-linear equations, is quite challenging. The new DD algorithms to be developed in this project will essentially contribute to a new generation software in Computational Electromagnetics.

Publications

Recent Preprints

• G. Of, O. Steinbach
• Is the one-equation coupling of finite and boundary element methods always stable
• INM-Bericht 2011/6, TU Graz
• pdf
• M. Kolmbauer
• Existence and uniqueness of eddy current problems in bounded and unbounded domains
• Numa-Report 2011-03, JKU Linz
• pdf

2012 - Refereed Journal Papers

• M. Kolmbauer
• Efficient solvers for multiharmonic eddy current optimal control problems with various constraints and their analysis
• IMA J. Numer. Math. published online, October 31, 2012
• doi preprint
• C. Pechstein, R. Scheichl
• Weighted Poincaré inequalities
• IMA J. Numer. Math. published online, October 15, 2012
• doi
• M. Kollmann, M. Kolmbauer, U. Langer, M. Wolfmayr, W. Zulehner
• A robust finite element solver for a multiharmonic parabolic optimal control problem
• Comput. Math. Appl. published online, 17 July 2012
• doi preprint
• M. Kolmbauer, U. Langer
• A robust preconditioned MinRes solver for time-periodic eddy current problems
• Comput. Methods Appl. Math. published online, 25 June 2012
• doi preprint
• M. Kollmann, M. Kolmbauer
• A preconditioned MinRes solver for time-periodic parabolic optimal control problems
• Numer. Linear Algebra Appl. published online, 27 April 2012
• doi preprint
• C. Pechstein
• Shape-explicit constants for some boundary integral operators
• Appl. Anal. 92(5), p. 949-974, 2013
• doi
• M. Kolmbauer, U. Langer
• A robust preconditioned MinRes solver for distributed time-periodic eddy current optimal control problems
• SIAM J. Sci. Comput. 34(6):B786–B809 (2012)
• doi preprint
• M. Kolmbauer
• A robust FEM-BEM MinRes solver for distributed multiharmonic eddy current optimal control problems in unbounded domains
• Electron. Trans. Numer. Anal. 39:231–252 (2012)
• pdf preprint
• G. Of, G.J. Rodin, M. Taus, O. Steinbach
• Coupling methods for interior penalty discontinuous Galerkin finite element methods and boundary element methods
• SIAM. J. Sci. Comput. 34(3):A1659–A1677 (2012)
• doi preprint

2012 - Refereed Proceedings Papers

• M. Kolmbauer, U. Langer
• A robust FEM-BEM Solver for time-harmonic eddy current problems
• In R. Bank, R. Kornhuber, O. Widlund (eds.), Domain Decomposition Methods in Science and Engineering XX, Lecture Notes in Computational Science and Engineering, Springer, 2012, to appear
• preprint
• O. Steinbach, M. Windisch
• Stable BETI methods in electromagnetics
• In R. Bank, R. Kornhuber, O. Widlund (eds.), Domain Decomposition Methods in Science and Engineering XX, Lecture Notes in Computational Science and Engineering, Springer, 2012, to appear
• C. Pechstein, M. Sarkis, R. Scheichl
• New theoretical coefficient robustness results for FETI-DP
• In R. Bank, R. Kornhuber, O. Widlund (eds.), Domain Decomposition Methods in Science and Engineering XX, Lecture Notes in Computational Science and Engineering, Springer, 2012, to appear

2011 - Refereed Journal Papers

• O. Steinbach
• A note on the stable coupling of finite and boundary elements
• SIAM J. Numer. Anal. 49(4):1521–1531 (2011)
• doi preprint
• C. Pechstein, R. Scheichl
• Analysis of FETI methods for multiscale PDEs - part II: interface variation
• Numer. Math. 118(3):485–529 (2011)
• doi preprint
• S. Engleder, O. Steinbach
• Boundary integral formulations for the forward problem in magnetic induction tomography
• Math. Methods Appl. Sci. 34(9):1144–1156 (2011)
• doi preprint
• O. Steinbach, M. Windisch
• Stable boundary element domain decomposition methods for the Helmholtz equation
• Numer. Math. 118(1):171-195 (2011)
• doi preprint

2011 - Refereed Proceedings Papers

• M. Kolmbauer, U. Langer
• A robust preconditioned MinRes solver for distributed time-periodic eddy current optimal control
• In R.E. Bank, W. Hackbusch, G. Wittum (eds.), Fast Solvers for Partial Differential Equations, volume 8 of Oberwolfach Reports, pp. 1561–1563, EMS Publishing House, 2011
• M. Kolmbauer, U. Langer
• A frequency-robust solver for the time-harmonic eddy current problem
• in: B. Michielsennd, J.-R. Poirier (eds.), Scientific Computing in Electrical Engineering SCEE 2010, volume 16 of Mathematics in Industry, pp. 97–105, Springer-Verlag, Berlin, 2011
• preprint

2010 - Refereed Journal Papers

• P. D. Ledger, S. Zaglmayr
• hp-Finite element simulation of three-dimensional eddy current problems on multiply connected domains
• Comput. Methods Appl. Mech. Engrg. 199(49–52):3386–3401 (2010)
• doi preprint
• D. M. Copeland, U. Langer
• Domain decomposition solvers for nonlinear multiharmonic finite element equations
• J. Numer. Math. 18(3):157–176 (2010)
• doi preprint

2010 - Refereed Proceedings Papers

• D. Copeland, M. Kolmbauer, U. Langer
• Domain Decomposition Solvers for Frequency-Domain Finite Element Equations
• Domain Decomposition Methods in Science and Engineering XIX (Y. Huang, R. Kornhuber, O. Widlund, and J. Xu, eds.), volume 78 of Lecture notes in Computational Science and Engineering, Springer-Verlag, Berlin, 2010, pp. 301–308
• preprint
• G. Of, O. Steinbach
• Coupled FE/BE formulations for the fluid-structure interaction
• Domain Decomposition Methods in Science and Engineering XIX (Y. Huang, R. Kornhuber, O. Widlund, and J. Xu, eds.), volume 78 of Lecture notes in Computational Science and Engineering, Springer-Verlag, Berlin, 2010, pp. 293–300
• preprint
• C. Pechstein, R. Scheichl
• Weighted Poincaré inequalities and applications in domain decomposition
• Domain Decomposition Methods in Science and Engineering XIX (Y. Huang, R. Kornhuber, O. Widlund, and J. Xu, eds.), volume 78 of Lecture Notes in Computational Science and Engineering, Springer-Verlag, Berlin, 2010, pp. 197–204
• preprint
• O. Steinbach, M. Windisch
• Robust boundary element domain decomposition solvers in acoustics
• Domain Decomposition Methods in Science and Engineering XIX (Y. Huang, R. Kornhuber, O. Widlund, and J. Xu, eds.), volume 78 of Lecture Notes in Computational Science and Engineering, Springer-Verlag, Berlin, 2010, pp. 277–284
• preprint
• C. Pechstein, R. Scheichl
• Robust FETI Solvers for Multiscale Elliptic PDEs
• Scientific Computing in Electrical Engineering (J. Roos and L.R.J. Costa, eds.), volume 14 of Mathematics in Industry, Springer, Berlin, 2010, pp. 421–428 (Proceedings of the SCEE 2008, Espoo, Finland)

2009 - Refereed Journal Papers

• G. Of, O. Steinbach
• The all-floating boundary element tearing and interconnecting method
• J. Numer. Math. 17(4):277–298 (2009)
• doi preprint
• C. Pechstein, R. Scheichl
• Scaling up through domain decomposition
• Appl. Anal. 88(10,11):1589–1608 (2009)
• doi preprint
• C. Pechstein
• Boundary element tearing and interconnecting methods in unbounded domains
• Appl. Num. Math. 59(11):2824–2842 (2009)
• doi
• O. Steinbach, M. Windisch
• Modified combined field integral equations for electromagnetic scattering
• SIAM J. Numer. Anal. 47(2):1149–1167 (2009)
• doi preprint
• G. Of, M. Kaltenbacher, O. Steinbach
• Fast multipole boundary element method for electrostatic field computations
• Compel 28(2):304–319 (2009)
• doi preprint
• D. M. Copeland
• Boundary-element-based finite element methods for Helmholtz and Maxwell equations on general polyhedral meshes
• Int. J. Appl. Math. Comput. Sci. 5(1):60–73 (2009)
• doi preprint

2009 - Refereed Proceedings Papers

• D. Copeland, U. Langer, D. Pusch
• From the boundary element method to local Tretz finite element methods on polyhedral meshes
• In M. Bercovier, M. J. Gander, R. Kornhuber, and O. Widlund, editors, Domain Decomposition Methods in Science and Engineering XVIII, Springer, Heidelberg, 2009, pp. 315–322 (Proceedings of the DD18, Jerusalem, Israel, 2008)
• preprint

2008 - Refereed Journal Papers

• C. Pechstein, R. Scheichl
• Analysis of FETI methods for multiscale PDEs
• Numer. Math. 111(2):293–333 (2008)
• doi preprint
• D. M. Copeland, J. Gopalakrishnan, J. E. Pasciak
• A mixed method for axisymmetric div-curl systems
• Math. Comp. 77(264):1941–1965 (2008)
• doi preprint
• S. Beuchler, T. Eibner, U. Langer
• Primal and dual interface concentrated iterative substructuring methods
• SIAM J. Num. Anal. 46(6):2818–2842 (2008)
• doi preprint
• S. Engleder, O. Steinbach
• Stabilized boundary element methods for exterior Helmholtz problems
• Numer. Math. 110(2):145–160 (2008)
• doi preprint
• G. Of
• An efficient algebraic multigrid preconditioner for a fast multipole boundary element method
• Computing 82(2–3):139–155 (2008)
• doi preprint
• U. Langer, C. Pechstein
• All-floating coupled data-sparse boundary and interface-concentrated finite element tearing and interconnecting methods
• Comput. Vis. Sci. 11(4–6):307–317 (2008)
• doi preprint

2008 - Refereed Proceedings Papers

• U. Langer
• Boundary and finite element domain decomposition methods
• In: Numerical Mathematics and Advanced Applications – Proceedings of the 7^th European Conference on Numerical Mathematics and Advanced Applications, Graz, Austria, September 2007 (ed. by K. Kunisch, O. Steinbach, G. Of), Springer, Heidelberg, 2008, pp. 15–26
• C. Pechstein
• BETI-DP methods in unbounded domains
• In: Numerical Mathematics and Advanced Applications – Proceedings of the 7^th European Conference on Numerical Mathematics and Advanced Applications, Graz, Austria, September 2007 (ed. by K. Kunisch, O. Steinbach, G. Of), Springer, Heidelberg, 2008, pp. 381–388
• G. Of
• The all-floating BETI method: numerical results
• In: Domain Decomposition Methods in Science and Engineering XVII (ed. by U. Langer, M. Discacciati, D. Keyes, O. Widlund, and W. Zulehner), vol. 60 of Lecture Notes in Compuational Science and Engineering, Springer, Berlin, Heidelberg, 2008, pp. 295–302
• preprint
• O. Steinbach
• Challenges and applications of boundary element domain decomposition methods
• In: Domain Decomposition Methods in Science and Engineering XVII (ed. by U. Langer, M. Discacciati, D. Keyes, O. Widlund, and W. Zulehner) Lecture Notes in Compuational Science and Engineering, Vol. 60, Springer, Berlin, Heidelberg, 2008 pp. 131–142
• preprint

Among the following publications are also some earlier relevant works by members and associate members of this FWF project.

2007 and Before - Refereed Journal Papers

• S. Engleder, O. Steinbach
• Modified boundary integral formulations for the Helmholtz equation
• J. Math. Anal. Appl. 331:396–407 (2007)
• doi preprint
• U. Langer, G. Of, O. Steinbach, W. Zulehner
• Inexact data-sparse boundary element tearing and interconnecting methods
• SIAM J. Sci. Comp. 29(1):290–314 (2007)
• doi preprint
• D. M. Copeland, J. E. Pasciak
• A least-squares method for axisymmetric div-curl systems
• Numer. Linear Algebra Appl., 13(9):733–752 (2006)
• doi
• U. Langer, C. Pechstein
• Coupled finite and boundary element tearing and interconnecting solvers for nonlinear potential problems
• ZAMM Z. Angew. Math. Mech. 86(12):915–931 (2006)
• doi
• C. Pechstein, B. Jüttler
• Monotonicity-preserving interproximation of B-H-curves
• J. Comput. Appl. Math. 196(1):45–57 (2006)
• doi preprint
• F. Bachinger, U. Langer, J. Schöberl
• Efficient solvers for nonlinear time-periodic eddy current problems
• Comput. Vis. Sci. 9(4):197–207 (2006)
• doi preprint
• F. Bachinger, U. Langer, J. Schöberl
• Numerical analysis of nonlinear multiharmonic eddy current problems
• Numer. Math. 100:593–616 (2005)
• doi preprint
• U. Langer, O. Steinbach
• Boundary element tearing and interconnecting methods
• Computing 71(3):205–228 (2003)
• doi
• preprint

2007 and Before - Refereed Proceedings Papers

• U. Langer, C. Pechstein
• Coupled FETI/BETI solvers for nonlinear potential problems in (un)bounded domains
• In: Scientific Computing in Electrical Engineering (ed. by G. Ciuprina and D. Ioan), Mathematics in Industry: The European Consortium for Mathematics in Industry, Vol. 11, Springer, Berlin, Heidelberg, 2007
• doi
• U. Langer, G. Of, O. Steinbach, W. Zulehner
• Inexact fast multipole BETI methods
• In: "Domain Decomposition Methods in Sciences and Engineering" (ed. by D.E. Keyes, O.B. Widlund), Lecture Notes in Computational Sciences and Engineering, Vol. 55, Springer, Berlin, Heidelberg, 2007, pp. 407–414
• preprint
• U. Langer, O. Steinbach
• Coupled boundary and finite element tearing and interconnecting methods
• In: "Domain Decomposition Methods in Sciences and Engineering" (ed. by R. Kornhuber, R. Hoppe, J. Periaux, O. Pironneau, O. Widlund, J. Xu), Lecture Notes in Computational Sciences and Engineering, Vol. 40, Springer, Heidelberg, 2004, pp. 83–97
• preprint

Books, Chapters in Books

• C. Pechstein
• Finite and Boundary Element Tearing and Interconnecting Solvers for Multiscale Problems
• Vol. 90 of Lecture Notes in Computational Science and Engineering, Springer-Verlag. 2013
• doi
• O. Steinbach
• Numerical Approximation Methods for Elliptic Boundary Value Problems – Finite and Boundary Elements
• Springer, New York 2008
•  
• S. Rjasanow, O. Steinbach
• The Fast Solution of Boundary Integral Equations – Mathematical and Analytical Techniques with Applications to Engineering
• Springer, New York 2007
•  
• U. Langer, O. Steinbach
• Coupled finite and boundary element domain decomposition methods
• In: "Boundary Element Analysis: Mathematical Aspects and Application", ed. by M. Schanz and O. Steinbach, Lecture Notes in Applied and Computational Mechanic, Vol. 29, Springer, Berlin, pp. 29–59 (2007)
•  

Editorial

• U. Langer, M. Schanz, O. Steinbach, W. L. Wendland (editors)
• Fast Boundary Element Methods in Engineering and Industrial Applications
• volume 63 of Lecture Notes in Applied and Compuational Mechanics, Springer-Verlag, Berlin, Heidelberg, 2012
•  
• U. Langer, O. Steinbach (editors)
• Analysis and Numerics of Boundary Integral Equations
• Mathematical Methods in the Applied Sciences, Vol. 31(17), 2008, (special issue dedicated to W.L. Wendland on the occasion of his 70th birthday)
•  
• K. Kunisch, G. Of, O. Steinbach (editors)
• Proceedings of ENUMATH 2007, Graz, September 2007
• Springer, Heidelberg, 2008
•  
• T. Apel, A. Böttcher, G. Haase, B. Heinrich, M. Jung, U. Langer, O. Steinbach, A. Meyer, A. Rösch (editors)
• Selected Papers from the 20th Chemnitz Finite Element Symposium
• Electronic Transactions on Numerical Analysis, Vol. 32 (2008), (special issue)
•  

PhD Theses

• M. Windisch
• Boundary Element Tearing and Interconnecting Methods for Acoustic and Electromagnetic Scattering
• Graz University of Technology, Graz, Austria, submitted Dezember 2010; printed in volume 11 of Monographic Series TU Graz: Computation in Engineering and Science, Verlag der Technischen Universität Graz, 2011
•  
• Sarah Engleder
• Boundary Element Methods for Eddy Current Transmission Problems
• Graz University of Technology, Graz, Austria, April 2011
•  
• C. Pechstein
• Finite and Boundary Element Tearing and Interconnecting Methods for Multiscale Elliptic Partial Differential Equations
• Johannes Kepler University, Linz, Austria, December 2008
• pdf
• S. Zaglmayr
• High Order Finite Element Methods for Electromagnetic Field Computations
• Johannes Kepler University, Linz, Austria, July 2006
• pdf
• D. M. Copeland
• Negative-norm least-squares methods for axisymmetric Maxwell equations
• Texas A&M University, May 2006
• pdf
• G. Of
• BETI domain decomposition methods with fast boundary element methods and applications
• University of Stuttgart, January 2006 (in german)
• pdf

Master Theses

• M. Kowalska (now: Wolfmayr)
• Shape optimization based on black box optimization
• master thesis, Johannes Kepler University, Linz, May 2010
• pdf
• M. Kolmbauer
• A multiharmonic solver for nonlinear parabolic problems
• master thesis, Johannes Kepler University, Linz, October 2009
• pdf
• M. Windisch
• Modified boundary integral equations for electromagnetic scattering problems
• master thesis, Graz University of Technology, March 2007 (in german)
• pdf
• S. Engleder
• Stabilized boundary integral equations for exterior boundary value problems of the Helmholtz Equation
• master thesis, Graz University of Technology, November 2006 (in german)
• pdf
• C. Pechstein
• Multigrid-Newton methods for nonlinear magnetostatic problems
• master thesis, Johannes Kepler University Linz, February 2004
• pdf

Technical reports

• C. Pechstein
• Analysis of dual and dual-primal tearing and interconnecting methods in unbounded domains
• SFB-Report 2007-15, Johannes Kepler University Linz, August 2007
• pdf
• U. Langer, A. Pohoata, O. Steinbach
• Dual-primal boundary element tearing and interconnecting methods
• Technical report 2005/6, Institute of Computational Mathematics, TU Graz, 2006
• pdf

Software

ParMax

We have been successively developing a code framework for simplifying and unifying the software connected with our research. The framework is in C++ using high standard template and object-oriented techniques. A great consultant is Joachim Schöberl, the founder of NGSolve.

Available:

• fast linear algebra (vectors, matrices, expression templates), interfaces to various LAPACK routines, and to the sparse direct solvers PARDISO, UMFPACK
• distributed meshes, interface to METIS
• abstract parallel framework, based on MPI
• generic FEM (in particular for H¹ and H(curl)), also high order
• interface to G. Of's BEM library
• BEM-based FEM
• basic multigrid, interface to Hypre's BoomerAMG
• multiharmonic solvers
• abstract FETI/BETI(-DP) methods
• applications to
  • non-linear magnetic field problems
  • non-linear elasticity
  • Stokes flow
  • optimal control

ParMax developers:

• C. Pechstein (JKU Linz, coordinator)
• C. Hofreither (JKU Linz)
• C. Augustin (Meduni Graz)
• M. Wolfmayr (JKU Linz)
• E. Karabelas (TU Graz)

Former developers:

• D. Copeland (RICAM Linz and Texas A&M University)
• M. Kolmbauer (JKU Linz)