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KARDOS - User"s Guide
(2002)
The adaptive finite element code {\sc Kardos} solves nonlinear parabolic systems of partial differential equations. It is applied to a wide range of problems from physics, chemistry, and engineering in one, two, or three space dimensions. The implementation is based on the programming language C. Adaptive finite element techniques are employed to provide solvers of optimal complexity. This implies a posteriori error estimation, local mesh refinement, and preconditioning of linear systems. Linearely implicit time integrators of {\em Rosenbrock} type allow for controlling the time steps adaptively and for solving nonlinear problems without using {\em Newton's} iterations. The program has proved to be robust and reliable. The user's guide explains all details a user of {\sc Kardos} has to consider: the description of the partial differential equations with their boundary and initial conditions, the triangulation of the domain, and the setting of parameters controlling the numerical algorithm. A couple of examples makes familiar to problems which were treated with {\sc Kardos}. We are extending this guide continuously. The latest version is available by network: {\begin{rawhtml} <A href="http://www.zib.de/Numerik/software/kardos/"> <i> Downloads.</i></a> \end{rawhtml}}
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.
Adaptive numerical methods in time and space are introduced and studied for linear poroelastic models in two and three space dimensions. We present equivalent models for linear poroelasticity and choose both the {\em displacement--pressure} and the {\em stress--pressure} formulation for our computations. Their discretizations are provided by means of linearly implicit schemes in time and linear finite elements in space. Our concept of adaptivity opens a way to a fast and reliable simulation of different loading cases defined by corresponding boundary conditions. We present some examples using our code {\sf Kardos} and show that the method works efficiently. In particular, it could be used in the simulation of some bone healing models.