R. Beck, P. Deuflhard, R. Hiptmair, R. H. W. Hoppe, and B. Wohlmuth. Adaptive multilevel methods for edge element discretizations of Maxwell's equations. Surveys Math. Indust., 8:271--312, 1999.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....including the nontrivial kernel of the curl operator and coupling of field components in three dimensions make the latter operator much more di#cult to handle. This realization led to intense research activity into suitable computational methods in three dimensions (e.g. edge elements [11, 16]) Although finite element methods are rich in theoretical tools and provide great geometric flexibility, finite di#erence methods and finite volume methods are simpler to implement and can still provide accurate descriptions of solutions for many realistic electromagnetic phenomena [6, 25, 44, ....

....on frequency domain formulations of Maxwell s equations. As such, the discussion that follows is limited to well posed PDE problems in the frequency domain. For results describing initial and boundary conditions for well posed PDE problems based on Maxwell s equations in the time domain, see [4, 11, 70, 73, 97]. 2.3.1 Frequency Domain Boundary Conditions First, consider the time harmonic PDE problem (2.12) specified in an 31 unbounded spatial domain # . Then, as x ##, provided that x E(x, #) and x H(x, #) are bounded and that E(x, #) xH(x,#) o (2.22) uniformly in all ....

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R. Beck, P. Deuflhard, R. Hiptmair, R. Hoppe, and B. Wohlmuth. Adaptive multilevel methods for edge element discretizations of Maxwell's equations. Surveys Math. Indust., 8:271--312, 1999.

....Thus, the Helmholtz decomposition followed by a careful discretization allows for the construction of a simpler preconditioner for a standard Lanczos type iteration. Others have chosen to discretize (1) or the corresponding time domain equations) first, and then manipulate the discrete equations [5, 3, 9], possibly with the view of designing a fast solver [17] In the present setting our modular approach yields a complete scheme for fast 3D simulation which is easy to understand and implement by scientists and engineers. 4 2 Formulating the Electromagnetic Problem With a time dependence c t, ....

R. Beck, P. Deuflhard, R. Hiptmair, R. Hoppe, and B. Wohlmuth. Adap- tive multilevel methods for edge element discretizations of maxwel]'s equations. 'uvve]s Math. Idust., 1999. to appear.

.... I believe that a succesful and competitive implementation of a fully automatic hp method must be based on a fully integrated approach where both the error estimation and mesh optimization are imbedded into a multigrid solver, in a similar way as this has been done for h adaptive methods 16 [5, 21]. Acknowledgment The work has been supported by Air Force under Contract F49620 98 1 0255. ....

R. Beck, P. Deuflhard, R. Hiptmair, R.H.W. Hoppe, and B. Wohlmuth, "Adaptive Multilevel Methods for Edge Element Discretizations of Maxwell's Equations", Surveys on Mathematics for Industry, 8, 271-312, 1999.

....To illustrate this, we report iteration counts for the case # = 10 in Tables 5.3 and 5.4. Clearly in this case the coarse grid needs to be fine enough for a good preconditioner to result. Heuristic arguments indicating that coarse mesh size H should not be taken larger than # # exist in literature [2]. Tables 5.3 and 5.4 indicate that taking H = # # may not be su#cient for lowest order elements. Finally we investigate if we can replace the subdomain solves by preconditioners on subdomains and still get a good preconditioner. Let B a be the matrix of the operator B a defined by B a (#) N ....

R. Beck, P. Deuflhard, R. Hiptmair, R. H. W. Hoppe, and B. Wohlmuth. Adaptive multilevel methods for edge element discretizations of Maxwell's equations. Surveys Math. Indust., 8:271--312, 1999. 16 JAYADEEP GOPALAKRISHNAN AND JOSEPH E. PASCIAK

....Coulomb gauge, finite volume, Krylov methods, mixed methods, preconditioning. 1 Introduction The need for calculating fast, accurate solutions of three dimensional electromagnetic equations arises in many important application areas including, among others, geophysical surveys and medical imaging [29, 32, 2]. Consequently, a lot of effort has recently been invested in finding appropriate numerical algorithms. However, while it is widely agreed that electromagnetic phenomena are generally governed by Maxwell s equations, the choice of numerical techniques to solve these equations depends on parameter ....

.... while it is widely agreed that electromagnetic phenomena are generally governed by Maxwell s equations, the choice of numerical techniques to solve these equations depends on parameter ranges and various other restrictive assumptions, and as such is to a significant degree application dependent [20, 32, 2]. The present article is motivated by remote sensing inverse problems, e.g. in geophysics, where one seeks to recover material properties especially conductivity in an isotropic but heterogeneous body, based on measurements of electric and magnetic fields on or near the earth s surface. The ....

[Article contains additional citation context not shown here]

R. Beck, P. Deuflhard, R. Hiptmair, R. Hoppe, and B. Wohlmuth. Adaptive multilevel methods for edge element discretizations of maxwell's equations. Surveys Math. Indust., 1999. to appear.

....kernel of the rot operator, have to be taken into account carefully. A geometric multilevel method was set up by R. Hiptmair in [11] the first time. An other approach was discussed in [1] For applications on the geometric multigrid technique in the function space H(rot; Omega Gamma we refer to [11, 3, 1, 16, 20]. An algebraic multigrid approach for the solution of (1) requires in addition to the available components of the geometric multigrid also a proper coarsening strategy. In spite of the fact that the FE matrix K e h is SPD, the classical approaches of [5, 6, 7, 8, 15, 18, 19, 22] and variants of ....

....for (5) in the case of nonconvex domains Omega or if the coefficient function has a jump to get a good approximation of the continuous solution. Further applications in nonlinear or time dependent problems are out of the scope of this paper, and we refer to the extensive literature, see [3, 16, 20]. 5 3 Construction of an AMG Method In this section the ingredients of an AMG method are recalled and especially an approach for edge elements is proposed. Thus we are concerned with the pure algebraic construction of a multilevel hierarchy of coarse matrices for K e h . Therefore it is ....

R. Beck, P. Deuflhard, R. Hiptmair, R. Hoppe, and B. Wohlmuth, Adaptive multilevel methods for edge element discretizations of Maxwell's equations, Surveys Math. Indust. 8 (1999), 271--312.

.... z and k 0 also satis es the divergence condition (9) This can be seen by the special choice v = 1=k z r v z 2 H 0 (curl; and inserting equation (8) into (7) Given a regular triangulation of , the structure of (7) 9) can be passed on to a discrete version by using edge elements ( 16] 4] [1]) for the transversal components and nodal elements for u z . Let V H 0 (curl; V z H 1 0 be the corresponding linear nite element spaces with bases 1 m and 1 p . 4 Here m is the number of interior edges and p is the number of inner points of the triangulation. We ....

Beck, R., Deuhard, P., Hiptmair, R., Hoppe, R. H. W., and Wohlmuth, B.: Adaptive Multilevel Methods for Edge Element Discretizations of Maxwell's Equations. Surv. Math. Ind. 9 (1999) 271312

....be possible to perform all steps of a simulation for each individual patient within a medical planning system [1] The purpose of our paper is to describe an optimization process based on a three dimensional nonlinear heat transfer model. Finite element solutions of the electromagnetic fields [2] are taken as input data. It is a rather difficult task to establish an appropriate physical model for the heat transport in the human body. Several approaches can be found in the literature (see e.g. 18, 10] The basis for our modelling is Pennes bio heat transfer equation which we equip with ....

R. Beck, P. Deuflhard, R. Hiptmair, B. Wolmuth, R.H.W. Hoppe, Adaptive Multilevel Methods for Edge Element Discretizations of Maxwell's Equations, Preprint SC 97--66, Konrad--Zuse--Zentrum fur Informationstechnik Berlin, Germany, 1997

....as the mesh size decreases. To overcome this drawback, R. Hiptmair proposed to modify the smoothing iteration by adding a smoothing step in the discrete potential space [11] Similarly, D. Arnold, R. Falk and R. Winther suggested a special block smoother that has the same effect [1] In [2], one can find some implementation issues for the multigrid algorithm proposed by R. Hiptmair and numerical results for the eddy current problem that gives an additional L 2 term after the time discretization. The parallelization of these or, more precisely, of appropriately modified multigrid ....

R. Beck, P. Deuflhard, R. Hiptmair, R. Hoppe, and B. Wohlmuth. Adaptive multilevel methods for edge element discretizations of Maxwell's equations. Surv. Mat. Ind., 8:271--312, 1999.

....as stream function, vorticity and vector potential, 25] Certain electromagnetic phenomena are known to be modelled by Maxwell s equations. Here, the space H(curl; Omega Gamma appears when linking the quantities electric and magnetic field, magnetic induction and flux density, see for example [4, 6, 24] and the references therein. For the numerical treatment of these equations, it is very helpfull to have at hand bases for the kernel of the curl operator and its orthogonal complement. Received by the editor December 9, 1998, revised version April 27, 1999. 1991 Mathematics Subject ....

....the sequential regularization WAVELET BASES IN H(div) AND H(curl) 25 method for the nonstationary incompressible Navier Stokes equations and also certain plate problems, see [1, 6] and the references therein. In the context of Finite Element discretizations this has recently been studied in [1, 2, 4, 24]. We want to use the discrete Hodge decompositions for developing the preconditioners. Then, we apply the following result which is well known (see [16] and the references therein) Theorem 5.1. 16] Let the wavelet system Psi be stable in H s( Omega Gamma , i.e, kfk 2 H s ( Omega Gamma ....

R. Beck, P. Deuflhard, R. Hiptmair, R.H.W. Hoppe, and B. Wohlmuth, Adaptive multilevel methods for edge element discretizations of Maxwell's equations, ZIB Berlin, Report SC-9766, 1997, to appear in Surveys of Mathematics in Industry.

.... consistently at re entrant corners [6, 8] Though all macroscopic electromagnetic phenomena are governed by Maxwell s equations and some additional material laws, there is a wide range of electromagnetic problem types which are tackled by specific theoretical and numerical approaches (see, e.g. [4, 19, 20]) In this paper we confine ourselves to fields varying on a slow time scale. By this we mean that the wavelengths of the fields are not substantially smaller than the diameter of the computational domain; a typical example is the calculation of eddy currents. In general we have to distinguish ....

....for a suitable Krylov subspace solver like conjugate gradients or residuals. It is based on separate V cycles in the nullspace of curl and in the Nedelec space by exploiting discrete potentials via a Helmholtz decomposition. Similar strategies for geometric multigrid have been proposed in [4, 5, 17]. The space of potentials is a subspace of the Nedelec space and spanned by linear Lagrange type basis functions. Within this subspace a Poisson problem has to be solved, thus an efficient application of algebraic mesh coarsening is not difficult here. As for the Nedelec space, coarsening will be ....

[Article contains additional citation context not shown here]

R. Beck, P. Deuflhard, R. Hiptmair, R.H.W. Hoppe, and B. Wohlmuth. Adaptive multilevel methods for edge element discretizations of Maxwell's equations. Surveys on Mathematics for Industry, 8, no. 3-4, pp. 271--312, 1999.

....way to solve the discrete eddy current equations [31] To apply a generic multigrid method we need several stacked finite element meshes and corresponding data structures. A hierarchy of nested meshes also emerges naturally, when using modern adaptive techniques, which rely on local refinement [10, 12, 20, 47]: After the error of the discrete solution is estimated on a certain mesh, a new mesh is created by splitting elements into smaller ones where the accuracy is found wanting. Multilevel data structures (cf. 9, 11, 37] are challenging and take a big effort to implement. Nevertheless, the pay off ....

R. Beck, P. Deuflhard, R. Hiptmair, R. Hoppe, and B. Wohlmuth, Adaptive multilevel methods for edge element discretizations of Maxwell's equations, Tech. Rep. SC 97--66, ZIB Berlin, 1997. To appear in Surveys for Mathematics in Industry.

....ends, at z = 0 and z = 1 (the absorbing boundary part A ) we impose absorbing boundary conditions. More precisely, we put up with rst order Silver M uller boundary conditions [11, 36, 40] In the transient case the electric eld E = E(t; x) inside b is governed by the equations (see, e.g. [3, 36]) 2 t 2 E t E curl 1 curl E = t j in b curl E n ( p t E n) n = 0 on A E n = 0 on D : 1.1) Here = x) and = x) stand for the dielectric constant and the magnetic permeability, respectively. Both are uniformly positive functions in L 1 ( ....

.... ; 2 L 1 ( b depend on the material parameters and the length of the timestep. Both and are uniformly positive, must not be negative. All sources are lumped into f 2 L 2 ( b We point out that problems of the form (1. 2) occur in eddy current computations, as well [3]. By the Lax Milgram lemma a unique solution of (1.2) exists. We assume that neither f nor ; depend on the angle . Then the same applies to E and the rotationally symmetric setting is perfect. Passing to cylindrical coordinates we obtain two dimensional boundary value problems for ....

R. Beck, P. Deuflhard, R. Hiptmair, R. Hoppe, and B. Wohlmuth, Adaptive multilevel methods for edge element discretizations of Maxwell's equations, Surveys on Mathematics for Industry, 8 (1999), pp. 271-312.

....systems are solved iteratively by multigrid algorithms. A special hybrid smoothing technique, which utilizes a Helmholtz decomposition of the elds, is implemented for coping with the null space of the curl operator. Without this measure, iterative solvers may render a poor convergence behaviour [Beck et al. 1999]. Three types of boundary conditions occur in the simulations. The metallic parts of the antennas are assumed to be ideal conductors, represented by a homogeneous Dirichlet condition for the tangential component of the electric eld. At the antenna junctions, an inhomogeneous Cauchy condition is ....

Beck, R., Deu hard, P., Hiptmair, R., Hoppe, R. H. W., and Wohlmuth, B. (1999). Adaptive multilevel methods for edge element discretizations of Maxwell's equations. Surv. Math. Ind., 8:271-312.

.... this grid patient as a so called coarse grid adaptive multilevel finite element methods are applied to solve both Maxwell s equations (in the radio wave regime) and the bioheat transfer equation (linear and non linear) Details of these rather sophisticated recent numerical algorithms are given in [3,4]. e) The compational results are displayed by means of modern visualization tools which also permit a flexible 3D interaction with the virtual models at each stage of the planning process. The paradigm underlying the treatment planning system is as follows: map the essential features of an ....

Beck R, Deuflhard P, Hiptmair R, Wohlmuth B, Hoppe RHW. Adaptive Multilevel Methods for Edge Element Discretizations of Maxwell's Equations. Surv. Math. Ind. 1999; 8: 271-312.

....grid. The stiffness matrix and load vector corresponding to (2.3) are computed using Gaussian quadrature of order 5. Interpolation of boundary values is of the same order. The linear systems of equations are approximately solved by means of a multigrid preconditioned conjugate gradient method [14, 15]. The iterations are terminated, when the Euclidean norm of the algebraic residual for the current iterate is less than 10 Gamma10 times the Euclidean norm of the vector on the right hand side. Thus, the truncation error j it can be neglected. In all cases the local error estimator (3.10) in ....

R. Beck, P. Deuflhard, R. Hiptmair, R. Hoppe, and B. Wohlmuth, Adaptive multilevel methods for edge element discretizations of Maxwell's equations, Tech. Rep. SC 97--66, ZIB Berlin, 1997. To appear in Surveys for Mathematics in Industry.

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Beck R., Deuflhard P., Hiptmair R., Hoppe R., and Wohlmuth B. (1997) Adaptive multilevel methods for edge element discretizations of Maxwell's equations. ZIB Preprint SC 97-66. To appear in Surveys for Mathematics in Industry . ADAPTIVE MULTILEVEL FEM AS DECISIVE TOOLS ... 421

....linear continuous finite element functions S h on T h . Thus for E h 2 ND h we obtain a discrete Helmholtz decomposition E h = E hp E hs ; E hp = grad Phi h with Phi h 2 S h ; where E hs denotes the solenoidal part of E h (for a detailed description we refer to Hiptmair [11] and Beck et al. [2]. This decomposition with directly accessible potentials Phi h is of crucial importance for the construction of efficient solvers for the arising linear systems. Linear System Solution and Multilevel Preconditioning. The variational formulation (7) yields a sparse linear equation system Au = b ....

....smoothers. As a basic solver we use the conjugate residual (CR) method, which is similar to the well known conjugate gradient algorithm, but adjusted to symmetric indefinite systems [10] For preconditioning we set up a hybrid smoothing procedure, whose prerequisites are analyzed in detail in [2, 3]. Its basic operations are Gauss Seidel sweeps both in the N ed elec space ND h , coping with the elliptic part of A, and in the nullspace. Within this framework, efficient transfer operators between field representations in S h and ND h are essential. If we represent a vector field E hp in the ....

[Article contains additional citation context not shown here]

R. Beck, P. Deuflhard, R. Hiptmair, R.H.W. Hoppe, and B. Wohlmuth. Adaptive multilevel methods for edge element discretizations of Maxwell's equations. ZIB Preprint SC 97-66. To appear in Surveys for Mathematics in Industry.

No context found.

R. Beck, P. Deuflhard, R. Hiptmair, R. H. W. Hoppe, and B. Wohlmuth. Adaptive multilevel methods for edge element discretizations of Maxwell's equations. Surveys Math. Indust., 8:271--312, 1999.

No context found.

B. Beck, P. Deu hard, R. Hiptmair, R. H. W. Hoppe and B. Wohlmuth, Adaptive multilevel methods for edge element discretizations of Maxwell's equations, Surveys Math. Indust. 8 (1999), no. 3-4, 271-312.

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