@misc{BornemannYserentant, author = {Bornemann, Folkmar A. and Yserentant, Harry}, title = {A Basic Norm Equivalence for the Theory of Multilevel Methods.}, doi = {10.1007/BF01388699}, number = {SC-92-01}, abstract = {Subspace decompositions of finite element spaces based on \$L2\$-like orthogonal projections play an important role for the construction and analysis of multigrid like iterative methods. Recently several authors proved the equivalence of the associated discrete norms with the \$H^1\$-norm. The present report gives an elementary, self-contained derivation of this result which is based on the use of \$ K\$-functionals known from the theory of interpolation spaces. {\bf Keywords:} multilevel methods, nonuniform meshes, optimal convergence rates. {\bf AMS(MOS) Subject classifications:} 65N55, 65N30, 65N50.}, language = {en} } @misc{OevermannKlein, author = {Oevermann, Michael and Klein, Rupert}, title = {A cartesian grid finite volume method for the solution of the Poisson equation with variable coefficients and embedded interfaces}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-8996}, number = {06-05}, abstract = {We present a finite volume method for the solution of the two-dimensional Poisson equation \$ \nabla\cdot( \beta( {\mbox{\boldmath \$x\$}}) \nabla u({\mbox{\boldmath \$x\$}})) = f(\mbox{\boldmath \$x\$}) \$ with variable, discontinuous coefficients and solution discontinuities on irregular domains. The method uses bilinear ansatz functions on Cartesian grids for the solution \$u({\mbox{\boldmath \$x\$})\$ resulting in a compact nine-point stencil. The resulting linear problem has been solved with a standard multigrid solver. Singularities associated with vanishing partial volumes of intersected grid cells or the dual bilinear ansatz itself are removed by a two-step asymptotic approach. The method achieves second order of accuracy in the \$L^\infty\$ and \$L^2\$ norm.}, language = {en} } @phdthesis{Schmidt, author = {Schmidt, Frank}, title = {A New Approach to Coupled Interior-Exterior Helmholtz-Type Problems: Theory and Algorithms}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-7128}, number = {02-45}, abstract = {The work presents a new approach to the numerical solution of time-harmonic and time-dependent scattering problems. We replace Sommerfeld's radiation condition valid for the Helmholtz equation by a more general concept called pole condition. The pole condition is based on the Laplace transform of the exterior solution and allows a characterization of outgoing waves. Both new insight into the analysis of scattering problems as well as new numerical algorithms are obtained.}, language = {en} } @misc{ZschiedrichKloseSchaedleetal., author = {Zschiedrich, Lin and Klose, Roland and Sch{\"a}dle, Achim and Schmidt, Frank}, title = {A new Finite Element realization of the Perfectly Matched Layer Method for Helmholtz scattering problems on polygonal domains in 2D}, doi = {10.1016/j.cam.2005.03.047}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-7662}, number = {03-44}, abstract = {In this paper we propose a new finite element realization of the Perfectly Matched Layer method (PML-method). Our approach allows to deal with arbitrary shaped polygonal domains and with certain types of inhomogeneous exterior domains. Among the covered inhomogeneities are open waveguide structures playing an essential role in integrated optics. We give a detailed insight to implementation aspects. Numerical examples show exponential convergence behavior to the exact solution with the thickness of the PML sponge layer.}, language = {en} } @misc{OevermannScharfenbergKlein, author = {Oevermann, Michael and Scharfenberg, Carsten and Klein, Rupert}, title = {A sharp interface finite volume method for elliptic equations on Cartesian grids}, issn = {1438-0064}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-10900}, number = {08-38}, abstract = {We present a second order sharp interface finite volume method for the solution of the three-dimensional poisson equation with variable coefficients on Cartesian grids. In particular, we focus on interface problems with discontinuities in the coefficient, the source term, the solution, and the fluxes across the interface. The method uses standard piecewiese trilinear finite elements for normal cells and a double piecewise trilinear ansatz for the solution on cells intersected by the interface resulting always in a compact 27-point stencil. Singularities associated with vanishing partial volumes of intersected grid cells are removed by a two-term asymptotic approach. In contrast to the 2D method presented by two of the authors in [M.~Oevermann, R.~Klein: A Cartesian grid finite volume method for elliptic equations with variable coefficients and embedded interfaces, J.~Comp.~Phys.~219 (2006)] we use a minimization technique to determine the unknown coefficients of the double trilinear ansatz. This simplifies the treatment of the different cut-cell types and avoids additional special operations for degenerated interface topologies. The resulting set of linear equations has been solved with a BiCGSTAB solver preconditioned with an algebraic multigrid. In various testcases -- including large coefficient ratios and non-smooth interfaces -- the method achieves second order of accuracy in the L_inf and L_2 norm.}, language = {en} } @misc{Bornemann, author = {Bornemann, Folkmar A.}, title = {A Sharpened Condition Number Estimate for the BPX Preconditioner of Elliptic Finite Element Problems on Highly Nonuniform Triangulations.}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-596}, number = {SC-91-09}, abstract = {In this paper it is shown that for highly nonuniformly refined triangulations the condition number of the BPX preconditioner for elliptic finite element problems grows at most linearly in the depth of refinement. This is achieved by viewing the computational available version of the BPX preconditioner as an abstract additive Schwarz method with exact solvers. {\bf AMS CLASSIFICATION:} 65F10, 65F35, 65N20, 65N30.}, language = {en} } @misc{BraessDeuflhardLipnikov, author = {Braess, Dietrich and Deuflhard, Peter and Lipnikov, Konstantin}, title = {A Subspace Cascadic Multigrid Method for Mortar Elements}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-3954}, number = {SC-99-07}, abstract = {A cascadic multigrid (CMG) method for elliptic problems with strong material jumps is proposed and analyzed. Non--matching grids at interfaces between subdomains are allowed and treated by mortar elements. The arising saddle point problems are solved by a subspace confined conjugate gradient method as smoother for the CMG. Details of algorithmic realization including adaptivity are elaborated. Numerical results illustrate the efficiency of this CMG algorithm.}, language = {en} } @misc{SchmidtFrieseZschiedrichetal., author = {Schmidt, Frank and Friese, Tilmann and Zschiedrich, Lin and Deuflhard, Peter}, title = {Adaptive Multigrid Methods for the Vectorial Maxwell Eigenvalue Problem for Optical Waveguide Design}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-6228}, number = {00-54}, abstract = {This paper has been motivated by the need for a fast robust adaptive multigrid method to solve the vectorial Maxwell eigenvalue problem arising from the design of optical chips. Our nonlinear multigrid methods are based on a previous method for the scalar Helmholtz equation, which must be modified to cope with the null space of the Maxwell operator due to the divergence condition. We present two different approaches. First, we present a multigrid algorithm based on an edge element discretization of time-harmonic Maxwell's equations, including the divergence condition. Second, an explicit elimination of longitudinal magnetic components leads to a nodal discretization known to avoid discrete \emph{spurious modes} also and a vectorial eigenvalue problem, for which we present a multigrid solver. Numerical examples show that the edge element discretization clearly outperforms the nodal element approach.}, language = {en} } @misc{DeuflhardSeebass, author = {Deuflhard, Peter and Seebass, Martin}, title = {Adaptive Multilevel FEM as Decisive Tools in the Clinical Cancer Therapy Hyperthermia}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-3739}, number = {SC-98-30}, abstract = {The paper surveys recent progress in a joint mathematical-medical project on cancer therapy planning. Within so-called regional hyperthermia the computational task is to tune a set of coupled radiofrequency antennas such that a carefully measured tumor is locally heated, but any outside hot spots are avoided. A mathematical model of the whole clinical system -- air, applicator with antennas, water bolus, individual patient body -- involves Maxwell's equations in inhomogeneous media and a parabolic bioheat transfer equation, which represents a simplified model of heat transfer in the human body (ignoring strong blood vessel heat transport). Both PDEs need to be computed fast and to medical reliability (!) on a workstation within a clinical environment. This requirement triggered a series of new algorithmic developments to be reported here, among which is an adaptive multilevel FEM for Maxwell's equations, which dominates the numerical simulation time. In total, however, the main bulk of computation time (see Table 3 in Section 4 below) still goes into segmentation -- a necessary preprocessing step in the construction a 3D virtual patient from the input of a stack of 2D computed tomograms (left out here).}, language = {en} } @misc{Beck, author = {Beck, Rudolf}, title = {Algebraic Multigrid by Component Splitting for Edge Elements on Simplicial Triangulations}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-4290}, number = {SC-99-40}, abstract = {Our focus is on Maxwell's equations in the low frequency range; two specific applications we aim at are time-stepping schemes for eddy current computations and the stationary double-curl equation for time-harmonic fields. We assume that the computational domain is discretized by triangles or tetrahedrons; for the finite element approximation we choose N\'{e}d\'{e}lec's \$H(curl)\$-conforming edge elements of the lowest order. For the solution of the arising linear equation systems we devise an algebraic multigrid preconditioner based on a spatial component splitting of the field. Mesh coarsening takes place in an auxiliary subspace, which is constructed with the aid of a nodal vector basis. Within this subspace coarse grids are created by exploiting the matrix graphs. Additionally, we have to cope with the kernel of the \$curl\$-operator, which comprises a considerable part of the spectral modes on the grid. Fortunately, the kernel modes are accessible via a discrete Helmholtz decomposition of the fields; they are smoothed by additional algebraic multigrid cycles. Numerical experiments are included in order to assess the efficacy of the proposed algorithms.}, language = {en} } @misc{LangWalter, author = {Lang, Jens and Walter, Artur}, title = {An Adaptive Discontinuous Finite Element Method for the Transport Equation.}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-579}, number = {SC-91-07}, abstract = {In this paper we introduce a discontinuous finite element method. In our approach, it is possible to combine the advantages of finite element and finite difference methods. The main ingredients are numerical flux approximation and local orthogonal basis functions. The scheme is defined on arbitrary triangulations and can be easily extended to nonlinear problems. Two different error indicators are derived. Especially the second one is closely connected to our approach and able to handle arbitrary variing flow directions. Numerical results are given for boundary value problems in two dimensions. They demonstrate the performance of the scheme, combined with the two error indicators. {\bf Key words:} neutron transport equation, discontinuous finite element, adaptive grid refinement. {\bf Subject classifications:} AMS(MOS) 65N30, 65M15.}, language = {en} } @misc{Bornemann, author = {Bornemann, Folkmar A.}, title = {An Adaptive Multilevel Approach to Parabolic Equations III.}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-519}, number = {SC-91-01}, abstract = {Part III of the paper is devoted to the construction of an adaptive FEM solver in two spatial dimensions, which is able to handle the singularly perturbed elliptic problems arising from discretization in time. The problems of error estimation and multilevel iterative solution of the linear systems - both uniformly well behaved with respect to the time step - can be solved simultaneously within the framework of preconditioning. A multilevel nodal basis preconditioner able to handle highly nonuniform meshes is derived. As a numerical example an application of the method to the bioheat-transfer equation is included. {\bf AMS CLASSIFICATION:} 65F10, 65F35, 65M50, 65M60, 65N30.}, language = {en} } @misc{DeuflhardLeinenYserentant, author = {Deuflhard, Peter and Leinen, P. and Yserentant, Harry}, title = {Concepts of an Adaptive Hierarchical Finite Element Code.}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-131}, number = {SC-88-05}, abstract = {The paper presents the mathematical concepts underlying the new adaptive finite element code KASKADE, which, in its present form, applies to linear scalar second-order 2-D elliptic problems on general domains. Starting point for the new development is the recent work on hierarchical finite element bases due to Yserentant (1986). It is shown that this approach permits a flexible balance between iterative solver, local error estimator, and local mesh refinement device - which are the main components of an adaptive PDE code. Without use of standard multigrid techniques, the same kind of computational complexity is achieved - independent of any uniformity restrictions on the applied meshes. In addition, the method is extremely simple and all computations are purely local - making the method particularly attractive in view of parallel computing. The algorithmic approach is illustrated by a well-known critical test problem. {\bf Keywords:} finite elements, hierarchical basis, adaptive mesh refinement, preconditioned conjugate gradient methods.}, language = {en} } @misc{Deuflhard, author = {Deuflhard, Peter}, title = {Differential Equations in Technology and Medicine. Computational Concepts, Adaptive Algorithms, and Virtual Labs}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-4223}, number = {SC-99-34}, abstract = {This series of lectures has been given to a class of mathematics postdocs at a European summer school on Computational Mathematics Driven by Industrial Applications in Martina Franca, Italy (organized by CIME). It deals with a variety of challenging real life problems selected from clinical cancer therapy, communication technology, polymer production, and pharmaceutical drug design. All of these problems from rather diverse application areas share two common features: (a) they have been modelled by various differential equations -- elliptic, parabolic, or Schr{\"o}dinger--type partial differential equations, countable ordinary diffential equations, or Hamiltonian systems, (b) their numerical solution has turned out to be real challenge to computational mathematics.}, language = {en} } @misc{SeebassBeckGellermannetal., author = {Seebass, Martin and Beck, Rudolf and Gellermann, Johanna and Nadobny, Jacek and Wust, Peter}, title = {Electromagnetic phased arrays for regional hyperthermia -- optimal frequency and antenna arrangement}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-5961}, number = {00-28}, abstract = {In this paper we investigate the effects of the three-dimensional arrangement of antennas and frequency on temperature distributions that can be achieved in regional hyperthermia using an electromagnetic phased array. We compare the results of power-based and temperature-based optimization. Thus we are able to explain the discrepancies between previous studies favouring more antenna rings on the one hand and more antennas per ring on the other hand. We analyze the sensitivity of the results with respect to changes in amplitudes and phases as well as patient position. This analysis can be used for different purposes. First, it provides additional criteria for selecting the optimal frequency. Second, it can be used for specifying the required phase and amplitude accuracy for a real phased array system. Furthermore, it may serve as a basis for technological developments in order to reduce both types of sensitivities described above.}, language = {en} } @misc{BurgerKloseSchaedleetal., author = {Burger, Sven and Klose, Roland and Sch{\"a}dle, Achim and Zschiedrich, Lin}, title = {HelmPole - A finite element solver for scattering problems on unbounded domains: Implementation based on PML}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-7609}, number = {03-38}, abstract = {The solution of scattering problems described by the Helmholtz equation on unbounded domains is of importance for a wide variety of applications, for example in electromagnetics and acoustics. An implementation of a solver for scattering problems based on the programming language Matlab is introduced. The solver relies on the finite-element-method and on the perfectly-matched-layer-method, which allows for the simulation of scattering problems on complex geometries surrounded by inhomogeneous exterior domains. This report gives a number of detailed examples and can be understood as a user manual to the freely accessible code of the solver HelmPole.}, language = {en} } @misc{Walter, author = {Walter, Artur}, title = {Improvement of Incomplete Factorizations by a Sparse Secant Method.}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-401}, number = {SC-90-12}, abstract = {In the present paper, the improvement of an incomplete factorization of a non-symmetric matrix A is discussed. Starting from the ideas of sparsity preserving quasi-Newton methods, an algorithm is developed which improves the approximation of A by the incomplete factorization maintaining the sparsity structure of the matrices. No renumbering of the unknowns or the admittance of additional fill-in is necessary. The linear convergence of the algorithm is proved under the assumption, that \$ L \$ and \$ U \$* have the same sparsity structure and an incomplete factorization with some reasonable approximation property exits. In combination with this algorithm, the method of incomplete factorization and its several modifications are applicable to a wider class of problems with improved convergence qualities. This is shown by a numerical example. {\bf Key Words:} non-symmetric linear system, sparse secant method, incomplete factorization. AMS(MOS) {\bf Subject Classifications:} 65F10, 65N20, 65N30.}, language = {en} } @misc{KornhuberRoitzsch, author = {Kornhuber, Ralf and Roitzsch, Rainer}, title = {On Adaptive Grid Refinement in the Presence of Internal or Boundary Layers.}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-230}, number = {SC-89-05}, abstract = {We propose an anisotropic refinement strategy which is specially designed for the efficient numerical resolution of internal and boundary layers. This strategy is based on the directed refinement of single triangles together with adaptive multilevel grid orientation. It is demonstrated by several numerical examples that compared to usual methods, the new anisotropic refinement ends up in more stable and more accurate solutions at much less computational cost. {\bf Keywords:} Adaptive finite elements, directed refinement, adaptive grid orientation, convection diffusion equation, internal and boundary layers.}, language = {en} } @misc{Yserentant, author = {Yserentant, Harry}, title = {Preconditioning Indefinite Discretization Matrices.}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-81}, number = {SC-87-06}, abstract = {The finite element discretization of many elliptic boundary value problems leads to linear systems with positive definite and symmetric coefficient matrices. Many efficient preconditioners are known for these systems. We show that these preconditioning matrices can be used also for the linear systems arising from boundary value problems which are potentially indefinite due to lower order terms in the partial differential equation. Our main tool is a careful algebraic analysis of the condition numbers and the spectra of perturbed matrices which are preconditioned by the same matrices as in the unperturbed case. {\bf Keywords: }Preconditioned conjugate gradient methods, finite elements. {\bf Subject Classification: } AMS(MOS):65F10, 65N20, 65N30.}, language = {en} } @misc{BankDupontYserentant, author = {Bank, Randolph and Dupont, Todd F. and Yserentant, Harry}, title = {The Hierarchical Basis Multigrid Method.}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-44}, number = {SC-87-02}, abstract = {We derive and analyze the hierarchical basis-multigrid method for solving discretizations of self-adjoint, elliptic boundary value problems using piecewise linear triangular finite elements. The method is analyzed as a block symmetric Gauß- Seidel iteration with inner iterations, but it is strongly related to 2-level methods, to the standard multigrid V-cycle, and to earlier Jacobi-like hierarchical basis methods. The method is very robust, and has a nearly optimal convergence rate and work estimate. It is especially well suited to difficult problems with rough solutions, discretized using highly nonuniform, adaptively refined meshes.}, language = {en} }