@misc{KornhuberRoitzsch1989, author = {Kornhuber, Ralf and Roitzsch, Rainer}, title = {Adaptive Finite-Elemet- Methoden f{\"u}r konvektionsdominierte Randwertprobleme bei partiellen Differentialgleichungen.}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-170}, number = {SC-88-09}, year = {1989}, abstract = {Ausgangspunkt bei der Behandlung konvektiv dominierter, elliptischer Probleme sind die bekannten hierarchischen Finite-Element-Methoden f{\"u}r den rein elliptischen Fall. Als stabile Erweiterung des Standard-Galerkin-Verfahrens wird das Stromlinien-Diffusions-Verfahren durch physikalische {\"U}berlegungen motiviert und kurz diskutiert. Anschließend zeigen wir, daß diese Methode erst in Verbindung mit einer hier erstmals vorgestellten lokalen Ausrichtung der Kanten wirksam eingesetzt werden kann. Zusammen mit einer ebenfalls neu entwickelten richtungsorientierten Verfeinerungsstrategie erh{\"a}lt man eine erheblich stabilere, genauere und schnellere Aufl{\"o}sung von Grenzschichten als mit herk{\"o}mmlichen Methoden.}, language = {de} } @misc{KornhuberWittum1992, author = {Kornhuber, Ralf and Wittum, Gabriel}, title = {Discretization and Iterative Solution of Convection Diffusion Equations.}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-762}, number = {SC-92-06}, year = {1992}, abstract = {We propose an extended box method which turns out to be a variant of standard finite element methods in the case of pure diffusion and an extension of backward differencing to irregular grids if only convective transport is present. Together with the adaptive orientation proposed in a recent paper and a streamline ordering of the unknowns, this discretization leads to a highly efficient adaptive method for the approximation of internal layers in the case of large local Peclet numbers.}, language = {en} } @misc{KornhuberRoitzsch1991, author = {Kornhuber, Ralf and Roitzsch, Rainer}, title = {Self Adaptive Computation of the Breakdown Voltage of Planar pn-Junctions with Multistep Field Plates.}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-525}, number = {SC-91-02}, year = {1991}, abstract = {The breakdown voltage highly depends on the electric field in the depletion area whose computation is the most time consuming part of the simulation. We present a self adaptive Finite Element Method which reduces dramatically the required computation time compared to usual Finite Difference Methods. A numerical example illustrates the efficiency and reliability of the algorithm.}, language = {en} } @misc{HoppeKornhuber1991, author = {Hoppe, Ronald H. W. and Kornhuber, Ralf}, title = {Multilevel Preconditioned CG-Iterations for Variational Inequalities.}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-561}, number = {SC-91-06}, year = {1991}, abstract = {We consider such variational inequalities which either describe obstacle problems or result from an implicit time discretization of moving boundary problems of two phase Stefan type. Based on a discretization in space by means of continuous, piecewise linear finite elements with respect to a nested hierarchy of triangulations, in both cases we use iterative processes consisting of inner and outer iterations. The outer iterations are either active set strategies or generalized Newton methods while the inner iterations are preconditioned cg- iterations with multilevel preconditioners.}, language = {en} } @misc{KornhuberRoitzsch1989, 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}, year = {1989}, 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{KornhuberRoitzsch1990, author = {Kornhuber, Ralf and Roitzsch, Rainer}, title = {Self Adaptive FEM Simulation of Reverse Biased pn-Junctions.}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-389}, number = {SC-90-10}, year = {1990}, abstract = {The potential distribution of reverse biased pn-junctions can be described by a double obstacle problem for the Laplacian. This problem is solved by a self adaptive Finite Element Method involving automatic termination criteria for the iterative solver, local error estimation and local mesh refinement. Special attention is paid to the efficient resolution of the geometries typically arising in semiconductor device simulation. The algorithm is applied to a reverse biased pn- junction with multi-step field plate and stop- electrode to illustrate its efficiency and reliability.}, language = {en} } @misc{Kornhuber1997, author = {Kornhuber, Ralf}, title = {Globally Convergent Multigrid Methods for Porous Medium Type Problems}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-3142}, number = {SC-97-45}, year = {1997}, abstract = {We consider the fast solution of large, piecewise smooth minimization problems as typically arising from the finite element discretization of porous media flow. For lack of smoothness, usual Newton multigrid methods cannot be applied. We propose a new approach based on a combination of convex minization with {\em constrained} Newton linearization. No regularization is involved. We show global convergence of the resulting monotone multigrid methods and give logarithmic upper bounds for the asymptotic convergence rates.}, language = {en} } @misc{Kornhuber1998, author = {Kornhuber, Ralf}, title = {Monotone Iterations for Elliptic Variational Inequalities}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-3539}, number = {SC-98-10}, year = {1998}, abstract = {A wide range of free boundary problems occurring in engineering andindustry can be rewritten as a minimization problem for astrictly convex, piecewise smooth but non--differentiable energy functional.The fast solution of related discretized problemsis a very delicate question, because usual Newton techniquescannot be applied. We propose a new approach based on convex minimization and constrained Newton type linearization. While convex minimization provides global convergence of the overall iteration, the subsequent constrained Newton type linearization is intended to accelerate the convergence speed. We present a general convergence theory and discuss several applications.}, language = {en} } @misc{RoitzschKornhuber1990, author = {Roitzsch, Rainer and Kornhuber, Ralf}, title = {BOXES - a Program to Generate Triangulations from a Rectangular Domain Description.}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-4738}, number = {TR-90-09}, year = {1990}, abstract = {BOXES computes a triangulation from a 2D domain description which consists of an arbitrary set of rectangles. Each rectangle may have attributes to control the triangulating process, define subdomain classes, or specify boundary conditions. The output of the program can be used as a coarse grid for KASKADE or one of its variants. Additional features are extensive checking of the user input, graphical display, and simple editing.}, language = {en} } @incollection{PolthierBobenkoHildebrandtetal.2014, author = {Polthier, Konrad and Bobenko, Alexander and Hildebrandt, Klaus and Kornhuber, Ralf and Tycowicz, Christoph von and Yserentant, Harry and Ziegler, G{\"u}nter M.}, title = {Geometry processing}, volume = {1}, booktitle = {MATHEON - Mathematics for Key Technologies}, publisher = {EMS Publishing House}, isbn = {978-3-03719-137-8}, doi = {10.4171/137}, pages = {341 -- 355}, year = {2014}, language = {en} } @misc{DeuflhardKornhuberSanderetal.2014, author = {Deuflhard, Peter and Kornhuber, Ralf and Sander, Oliver and Schiela, Anton and Weiser, Martin}, title = {Mathematics cures virtual patients}, volume = {1}, journal = {MATHEON-Mathematics for Key Technologies}, editor = {Deuflhard, Peter and Gr{\"o}tschel, Martin and H{\"o}mberg, Dietmar and Horst, Ulrich and Kramer, J{\"u}rg and Mehrmann, Volker and Polthier, Konrad and Schmidt, Frank and Sch{\"u}tte, Christof and Skutella, Martin and Sprekels, J{\"u}rgen}, publisher = {European Mathematical Society}, pages = {7 -- 25}, year = {2014}, language = {en} } @article{GelssIssagaliKornhuber2024, author = {Gelß, Patrick and Issagali, Aizhan and Kornhuber, Ralf}, title = {Fredholm integral equations for function approximation and the training of neural networks}, volume = {6}, journal = {SIAM Journal on Mathematics of Data Science}, number = {4}, doi = {10.1137/23M156642X}, year = {2024}, language = {en} }