Differentiability of parabolic semi-flows in L"p-spaces and inertial manifolds

We consider the semi-flow defined by semi-linear parabolic equations in L"p-spaces and study the differentiable dependence on the initial data. Together with a spectral gap condition this implies existence of inertial manifolds in arbitrary space dimensions. The spectral gap condition is satisfied by the Laplace-Beltrami operator for a class of manifolds. The simplest examples are products of spheres. (orig.)Available from TIB Hannover: RR 1606(99-23) / FIZ - Fachinformationszzentrum Karlsruhe / TIB - Technische InformationsbibliothekSIGLEDEGerman

Convergence of the Neumann Laplacians on shrinking domains

Available from TIB Hannover: RR 1606(99-34) / FIZ - Fachinformationszzentrum Karlsruhe / TIB - Technische InformationsbibliothekSIGLEDEGerman

A new domain decomposition method for convection-dominated problems

In this paper, we present a new domain decomposition method for solving convection-diffusion problems with dominant convection. We combine the sequential algorithm with a parallel strategy to adapt to the special properties of the equations. The sequential algorithm is used in the downwind direction while the parallel algorithm is applied in the crosswind direction. In both algorithms, an overlapping domain decomposition process is introduced. In each patch, we solve a local convection-dominated problem with artificial boundary conditions by the streamline diffusion finite element method. A globally continuous discrete solution is constructed from all the patch-wise solutions. It is proven that this global solution converges to the exact solution with an order of O(h"3"/"2) in the L"2-norm as long as the overlapping width is kept to O(h"3"/"4 vertical stroke log h vertical stroke) in the crosswind direction and to O(h vertical stroke log h vertical stroke) in the downwind direction. The L_2-error of order O(h_2) can be obtained and the overlapping width in the crosswind direction can be reduced to O(h vertical stroke log h vertical stroke) if quadrilateral elements are used and oriented in the streamline and crosswind directions. It is emphasized that we do not require the patch size to be sufficiently small and do not iterate the whole process as usually done in other domain decomposition approaches. (orig.)Available from TIB Hannover: RR 1606(95-38) / FIZ - Fachinformationszzentrum Karlsruhe / TIB - Technische InformationsbibliothekSIGLEDEGerman

How accurate is the streamline diffusion finite element method?

We investigate the optimal accuracy of the streamline diffusion finite element method applied to convection-dominated problems. For linear/bilinear elements the theoretical order of convergence given in the literature is either O(h"3"/"2) for quasi-uniform meshes or O(h"2) for some uniform meshes. The determination of the optimal order in general was an open problem. By studying a special type of meshes, it is shown that the streamline diffusion method may actually converge with any order within this range depending on the characterization of the meshes. (orig.)SIGLEAvailable from TIB Hannover: RR 1606(95-22) / FIZ - Fachinformationszzentrum Karlsruhe / TIB - Technische InformationsbibliothekDEGerman

Stability of a resonant system of conservation laws modeling polymer flow

We prove L"1 uniqueness and stability for a resonant 2 x 2 system of conservation laws that arise as a model for two phase polymer flow in porous media. The analysis uses the equivalence of the Eulerian and Lagrangian formulation of this system, and the results are first established for an auxiliary scalar equation. Our methods are based on front tracking approximations for the auxiliary equation, and the Kruzkov entropy condition for scalar conservation laws. (orig.)SIGLEAvailable from TIB Hannover: RR 1606(99-16) / FIZ - Fachinformationszzentrum Karlsruhe / TIB - Technische InformationsbibliothekDEGerman

How accurate is the streamline diffusion finite element method?

We investigate the optimal accuracy of the streamline diffusion finite element method applied to convection-dominated problems. For linear/bilinear elements the theoretical order of convergence given in the literature is either O(h"3"/"2) for quasi-uniform meshes or O(h"2) for some uniform meshes. The determination of the optimal order in general was an open problem. By studying a special type of meshes, it is shown that the streamline diffusion method may actually converge with any order within this range depending on the characterization of the meshes. (orig.)SIGLEAvailable from TIB Hannover: RR 1606(95-22) / FIZ - Fachinformationszzentrum Karlsruhe / TIB - Technische InformationsbibliothekDEGerman

Analysis of a domain-splitting method for nonstationary convection-diffusion problems

We give an analysis of a parallel solution method for non-stationary convection-diffusion problems. The discretization is by the streamline diffusion finite element method combined with a blockwise implicit time stepping on an overlapping domain decomposition. The missing interior boundary values are predicted by extrapolation in time. This approach was first proposed and analyzed elsewhere for strongly parabolic problems for which it has satisfactory stability and approximation properties. Here, we extend these results to convection-diffusion problems with dominant transport where the necessary numerical damping is introduced by streamline diffustion. The argument is entirely based on energy techniques and applies to genereal finite element discretizations. (orig./AKF)Available from TIB Hannover: RR 1606(94-16) / FIZ - Fachinformationszzentrum Karlsruhe / TIB - Technische InformationsbibliothekSIGLEDEGerman

Trends in processor technology and their impact on Numerics for PDE's

Processor technology is dramatically advancing and promises enormous improvements in processing data for the next decade. On the other hand, much lower advances in moving data are expected such that the efficiency of many numerical software tools for Partial Differential Equations are restricted by the cost for memory access. We demonstrate how data locality and pipelining can achieve a significant percentage of the available huge computing power, and we explain the influence of processor technology on recent and future numerical simulation tools. Exemplarily, we describe hardware-related concepts for adaptive error control, multigrid/domain decomposition schemes and incompressible flow solvers and discuss their numerical and computational characteristics. (orig.)Available from TIB Hannover: RR 1606(99-31) / FIZ - Fachinformationszzentrum Karlsruhe / TIB - Technische InformationsbibliothekSIGLEDEGerman

Asymptotic analysis of the laminar viscous flow over a porous bed

We consider the laminar viscous channel flow over a porous surface. The size of the pores is much smaller than the size of the channel and it is important to determine the effective boundary conditions at the porous surface. We study the corresponding boundary layers, and, by a rigorous asymptotic expansion, we obtain Saffman's modification of the interface condition observed by Beavers and Joseph. The effective coefficient in the law is determined through an auxiliary boundary-layer type problem, whose computational and modeling aspects are discussed in detail. Furthermore, the approximation errors for the velocity and for the effective mass flow are given as powers of the characteristic pore size #epsilon#. Finally, we give the interface condition linking the effective pressure fields in the porous medium and in the channel, and determine the jump of the effective pressures explicitly. (orig.)SIGLEAvailable from TIB Hannover: RR 1606(99-33) / FIZ - Fachinformationszzentrum Karlsruhe / TIB - Technische InformationsbibliothekDEGerman

Remarks on quenching

SIGLEAvailable from TIB Hannover: RR 1606(93-15) / FIZ - Fachinformationszzentrum Karlsruhe / TIB - Technische InformationsbibliothekDEGerman