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On Multigrid Methods for Solving Electromagnetic Scattering Problems On Multigrid Methods for Solving Electromagnetic Scattering Problems Dissertation zur Erlangung des akademischen Grades eines Doktor der Ingenieurwissenschaften (Dr.-Ing.) der Technischen Fakultat¨ der Christian-Albrechts-Universitat¨ zu Kiel vorgelegt von Simona Gheorghe 2005 1. Gutachter: Prof. Dr.-Ing. L. Klinkenbusch 2. Gutachter: Prof. Dr. U. van Rienen Datum der mundliche¨ Prufung:¨ 20. Jan. 2006 Contents 1 Introductory remarks 3 1.1 General introduction . 3 1.2 Maxwell’s equations . 6 1.3 Boundary conditions . 7 1.3.1 Sommerfeld’s radiation condition . 9 1.4 Scattering problem (Model Problem I) . 10 1.5 Discontinuity in a parallel-plate waveguide (Model Problem II) . 11 1.6 Absorbing-boundary conditions . 12 1.6.1 Global radiation conditions . 13 1.6.2 Local radiation conditions . 18 1.7 Summary . 19 2 Coupling of FEM-BEM 21 2.1 Introduction . 21 2.2 Finite element formulation . 21 2.2.1 Discretization . 26 2.3 Boundary-element formulation . 28 3 4 CONTENTS 2.4 Coupling . 32 3 Iterative solvers for sparse matrices 35 3.1 Introduction . 35 3.2 Classical iterative methods . 36 3.3 Krylov subspace methods . 37 3.3.1 General projection methods . 37 3.3.2 Krylov subspace methods . 39 3.4 Preconditioning . 40 3.4.1 Matrix-based preconditioners . 41 3.4.2 Operator-based preconditioners . 42 3.5 Multigrid . 43 3.5.1 Full Multigrid . 47 4 Numerical results 49 4.1 Coupling between FEM and local/global boundary conditions . 49 4.1.1 Model problem I . 50 4.1.2 Model problem II . 63 4.2 Multigrid . 64 4.2.1 Theoretical considerations regarding the classical multi- grid behavior in the case of an indefinite problem . 64 4.2.2 Model problem I . 67 4.2.3 Model problem II . 75 CONTENTS 1 4.2.4 Operator-based preconditioners, in combination with multigrid . 78 4.3 Remarks on performance comparison . 84 5 Summary and conclusions 87 Appendices 88 A Scattering from an infinite circular cylinder 89 A.1 Perfect conductor . 89 A.2 Dielectric cylinder . 91 B References from functional analysis 93 List of symbols 97 Acknowledgements 103 Bibliography 105 2 CONTENTS Chapter 1 Introductory remarks 1.1 General introduction A large class of two-dimensional electromagnetic problems, among them the scat- tering of time-harmonic electromagnetic waves and their propagation in wave- guides with discontinuities is governed by the Helmholtz operator with open bound- aries. In order to solve the underlying second-order elliptic partial di®erential equation (PDE) numerically, the Finite-Element Method (FEM) has been suc- cessfully used and described in the engineering literature over the years. As the solutions of the corresponding boundary value problem must satisfy a Sommerfeld-type boundary condition at infinity, ensuring thus their uniqueness, several combinations between Finite Element and other methods which deliver a suitable boundary condition to be incorporated into the discretization have been proposed. An extensive survey of existing work on non-reflecting boundary con- ditions can be found in [17]. A classification in terms of the nature of the resulting boundary conditions (local and global) also reflects in the structure of the matri- ces that arise from this discretization: local boundary conditions, like Bayliss- Turkel, preserve the sparse character of the system matrix, while global boundary conditions that arise from the combination with boundary elements, Dirichlet-to- Neumann mapping or eigenfunctions expansions lead to full submatrices corre- 3 4 Chapter 1. Introductory remarks sponding to the boundary nodes. A short review of some of the most important methods, along with numeri- cal issues related to them is given in Chapters 1 and 2. Section 1.6 deals with the basic ideas behind the Bayliss-Turkel boundary conditions, described in [3], with the Dirichlet-to-Neumann mapping, whose development as a tool for solv- ing boundary value problems on truncated domains is due to J. B. Keller and D. Givoli cf. [18], [19] and with the eigenfunctions expansion method. In Chapter 2 we give a more detailed review of the Finite Element Method, the main problems characteristic to its application to the Helmholtz equation, as well as its combi- nation with the boundary element method, for which we mention some important results that serve to proof the existence and uniqueness of the solution. While the question of existence and uniqueness of solutions of Helmholtz problems is ad- dressed in some classical books as [8] or [9], its finite element discretization and some problems related to it, as the pollution e®ect for increasing wave numbers, have been studied in [28, 29, 30]. The problem of existence and uniqueness of the solution of Finite Element Method-Boundary Element Method (FEM-BEM) coupled problems for acoustic and electromagnetic scattering is treated in [27]. As described in Chapter 3, e±cient solution algorithms for the Helmholtz problem with open boundaries have been developed, among them direct solvers and those ones based on iterative methods (classical as well as Krylov subspace), used as stand-alone solvers or preconditioners. A review of the main numerical problems encountered when dealing with the Helmholtz problem is presented in [49], reflecting the amount of research that has been done for it. A study of Incom- plete LU (ILU) and Generalized Minimum Residual (GMRES ) as precondition- ers for the solution of the Helmholtz problem by FEM is to be found in [32], while other preconditioners have been developed especially for the Helmholtz operator, among them the analytic ILU preconditioner [15], the SoV preconditioner based on the separation of variables [42], as well as a class of preconditioners based on the discretization of the Laplacian, developed in 1983 by Bayliss, Goldstein and Turkel [2] and generalized in 2001 by Laird [37] and then in 2004, yielding 1.1. General introduction 5 the complex shifted Laplace preconditioner (CSL) whose e±ciency in combina- tion with di®erent Krylov subspace methods has been evaluated in [13]. A further improvement, consisting in the approximation of this preconditioner by means of multigrid methods, has been developed, by the same authors, in [12]. Multigrid solvers have been also extensively used [21, 20, 39], as their e±ciency for elliptic problems has been proved. Nevertheless, their application to the Helm– holtz equation poses some problems that sometimes lead to slow convergence or even divergence, when standard multigrid (that is, with linear smoothers) is used as a stand-alone solver, as shown in [10, 50] or other extensive numerical stud- ies, like that one in [33]. Modification of the standard multigrid algorithm, by identifying the levels where these problems appear and by using a Krylov-type smoother for them, instead of the classical Jacobi or Gauss-Seidel smoothers, as well as an ”outer-iteration” [11] or the use of a grid-dependent eigenvalue shift, in combination with under-interpolation [50] have been proposed. Another solution, meant to overcome the standard multigrid di±culties is the wave-ray multigrid method, which has been proposed by Brandt et al. [6]. Finally, a fast solver for exterior Helmholtz problems based on imbedding methods has been developed in [14]. The main contribution of this thesis, presented in Chapter 4, lies in numerical studies whose aim is to determine whether classical multigrid is well-suited for the solution of the considered problems. As our interest is to find optimal algo- rithms for the above mentioned problems, in the low frequency domain, we have studied in this chapter the applicability of standard geometric multigrid to those problems, as well as the use of some remedies that have been proposed in the lit- erature. Among them, the performance of multigrid accelerated Krylov subspace methods (mostly BICGSTAB) and the operator-based ”shifted-Laplace” precon- ditioner, in comparison to classical and full-multigrid, in terms of computation time and convergence have been studied. Based on these numerical studies, we present the main conclusions regarding the feasibility of multigrid methods for the considered problems in Chapter 5. 6 Chapter 1. Introductory remarks 1.2 Maxwell's equations Whenever dealing with electromagnetic field problems the fundamental equations are Maxwell’s equations. We will take their di®erential form as starting point: @B(x; t) E(x; t) = (Faraday’s law) 8 r £ ¡ @t > @D(x; t) > H(x; t) = + J(x; t) (Maxwell-Ampere law) (1.1) > r £ @t <> D(x; t) = ½(x; t) (Gauss’s law) r ¢ > B(x; t) = 0 > r ¢ > where:>E and H are the electric and magnetic field intensities, D and B are the elec- tric and magnetic flux densities and J and ½ are the electric current and electric charge densities. If the medium is non-dispersive, linear, isotropic and inhomoge- neous, the constitutive relations are written as: D(x; t) = " (x) E(x; t) (1.2) B(x; t) = ¹ (x) H(x; t) (1.3) J(x; t) = σe (x) E(x; t) (1.4) where " , ¹ and σe denote, respectively, the permittivity, permeability and con- ductivity of the medium under consideration. In the following we shall restrict to time-harmonic electromagnetic fields varying with an angular frequency ! = 2¼ f rad/sec. In this case all the above-mentioned fields have the following repre- sentation: j!t F (x; t) = Re F0 (x) e t R+; (1.5) h i 8 2 where the complex vector F(x) is referred to as the field phasor. Assuming that there are no charges in the medium (½ = 0), the divergence-free fields E0 and H0 will then be solutions of the ”reduced” Maxwell’s equations: r E (x) = j!¹H (x) £ 0 ¡ 0 (1.6) ( r H0(x) = j!" f E0(x) £ where σe + j!" = j!" f . For the rest of the thesis, we’ll refer only to phasor fields and for simplicity we 1.3.
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