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Adaptive Multilevel Solution of Nonlinear Parabolic PDE Systems. Theory, Algorithm, and Applications
(1999)

This monograph has been written to illustrate the interlocking of theory, algorithm, and application in developing solution techniques for complex PDE systems. A deep theoretical understanding is necessary to produce a powerful idea leading to a successful algorithm. Efficient and robust implementation is the key to make the algorithm perform satisfactorily. The extra insight obtained by solving real--life problems brings out the structure of the method more clearly and suggests often ways to improve the numerical algorithm. It is my intention to impart the beauty and complexity found in both the theoretical investigation of the adaptive algorithm proposed here, i.e., the coupling of Rosenbrock methods in time and multilevel finite elements in space, and its realization. I hope that this method will find many more interesting applications.

A new seasonal energy storage for thermal solar systems has been developed on the basis of an adsorption-desorption process. Design and optimization of this storage will be supported by numerical simulations of heat and mass transfer with KARDOS. This paper focuses on the unsteady heat transfer during the major operating step of energetic discharge of the storage, which is characterized by conductive heat transfer in the fixed bed and a strong heat source caused by the adsorption enthalpy. Results are interpreted concerning the influence of variations in the parameter set. The method of implementation of the differential equation will be shown as well as the post-processing and gridwriting programs.

An Adaptive Finite Element Method for Convection-Diffusion Problems by Interpolation Techniques.
(1991)

For adaptive solution of convection- difussion problems with the streamline-diffusion finite element method, an error estimator based on interpolation techniques is developed. It can be shown that for correctness of this error estimator a restriction of the maximum angle is to be sufficient. Compared to usual methods, the adaptive process leads to more accurate solutions at much less computational cost. Numerical tests are enclosed. {\bf Keywords: } Adaptive finite elements, convection- diffusion equation, internal and boundary layers, streamline-diffusion. {\bf Subject Classifications:} AMS(MOS): 65N15, 65N30

Dynamical process simulation of complex real-life problems often requires the use of modern algorithms, which automatically adapt both the time and space discretization in order to get error-controlled approximations of the solution. In this paper, a combination of linearly implicit time integrators of Rosenbrock type and adaptive multilevel finite elements based on a posteriori error estimates is presented. This approach has proven to work quite satisfactorily for a wide range of challenging practical problems. We show the performance of our adaptive method for two applications that arise in the study of flame balls and brine transport in porous media.

A software package for the adaptive solution of time--dependent reaction--diffusion systems and linear elliptic systems in one space dimension is presented. The used algorithm is based on fundamental arguments in J.~Lang, A.~Walter: {\it A Finite Element Method Adaptive in Space and Time for Nonlinear Reaction--Diffusion Systems.} IMPACT of Computing in Science and Engineering, 4, p.~269--314 (1992). Here, only brief outlines of the algorithm are given. This software package is based on the KASKADE toolbox B.~Erdmann, J.~Lang, R.~Roitzsch: {\it KASKADE -- Manual.} To appear as Technical Report TR 93--5, Konrad--Zuse--Zentrum (ZIB) (1993).

The KASKADE toolbox defines an interface to a set of C subroutines which can be used to implement adaptive multilevel Finite Element Methods solving systems of elliptic equations in two and three space dimensions. The manual contains the description of the data structures and subroutines. The main modules of the toolbox are a runtime environment, triangulation and node handling, assembling, direct and iterative solvers for the linear systems, error estimators, refinement strategies, and graphic utilities. Additionally, we included appendices on the basic command language interface, on file formats, and on the definition of the partial differential equations which can be solved. The software is available on the ZIB ftp--server {\tt elib} in the directory {\tt pub/kaskade}. TR 93--5 supersedes TR 89--4 and TR 89--05.

One important step in the fabrication of silicon-based integrated circuits is the creation of semiconducting areas by diffusion of dopant impurities into silicon. Complex models have been developed to investigate the redistribution of dopants and point defects. In general, numerical analysis of the resulting PDEs is the central tool to assess the modelling process. We present an adaptive approach which is able to judge the quality of the numerical approximation and which provides an automatic mesh improvement. Using linearly implicit methods in time and multilevel finite elements in space, we are able to integrate efficiently the arising reaction-drift-diffusion equations with high accuracy. Two different diffusion processes of practical interest are simulated.

A Finite Element Method Adaptive in Space and Time for Nonlinear Reaction-Diffusion- Systems.
(1992)

Large scale combustion simulations show the need for adaptive methods. First, to save computation time and mainly to resolve local and instationary phenomena. In contrast to the widespread method of lines, we look at the reaction- diffusion equations as an abstract Cauchy problem in an appropriate Hilbert space. This means, we first discretize in time, assuming the space problems solved up to a prescribed tolerance. So, we are able to control the space and time error separately in an adaptive approach. The time discretization is done by several adaptive Runge-Kutta methods whereas for the space discretization a finite element method is used. The different behaviour of the proposed approaches are demonstrated on many fundamental examples from ecology, flame propagation, electrodynamics and combustion theory. {\bf Keywords:} initial boundary value problem, Rothe- method, adaptive Runge-Kutta method, finite elements, mesh refinement. {\bf AMS CLASSIFICATION:} 65J15, 65M30, 65M50.

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.

Dynamical simulation of industrially relevant processes strongly advises the use of algorithms, which are {\em adaptive} both in time and in space discretization. The paper presents two alternatives: (a) a fully adaptive method of lines approach, which is based on finite difference methods and essentially applicable to 1D problems; (b) a fully adaptive Rothe method, which is based on a fast multilevel finite element method and applicable to 1D up to 3D.