Research Group of Prof. Dr. M. Griebel
Institute for Numerical Simulation
Title Nonlinear multigrid methods for the numerical simulation of coupled systems
Participants Thomas Gerstner Michael Griebel
Keywords PDEs, multigrid
Description This work was part of the project B7 "Surface flow in the continental crust: Mechanisms of infiltration and interactions" within the collaborate research center SFB 350 "Interactions between and Modelling of Continental Geosystems".

The main point of interest of this project is the modelling and numerical simulation of mechanisms of infiltration and interaction of surface flow in the continental crust. Starting point is a system of differential equations coupling the unknowns matrix-translation, velocity, temperature and concentration. It describes the two dimensional physical processes in saturated porous media.

The model has to be extended to cover also the case of flow near the surface. Therefore the Darcy law which models the fluid-matrix coupling has to be substituted by the Richardson law in order to allow unsaturated flow, as it appears in regions near the surface. Special attention is given to the influence of the temperature on the matrix-fluid coupling in the unsaturated regions, the draining of the soil, the irreversible local increasing of permeability due to thermical changes and the expansion of the soil in dependence from the local moisture percentage. Systematic studies of the nonlinear physical interplay of the components matrix, fluid and temperature in the unsaturated regions have to be carried out. Another aim of the project is the reproduction and analysis of experimentally verified test problems.

Moreover, the numerical model has to be extended to three dimensions - resulting in huge storage and computing time requirements. After linearisation the equations of the model are solved efficiently by the use of multilevel methods. Especially robust multilevel methods - utilizing for example matrix dependent prolongations, ILU smoothers and algebraic coarsening (AMG-methods) - have to be used. A further increase of the efficiency is due to parallelization. The plan is to implement a domain decomposition method together with a multilevel solver on each subdomain on a cluster of workstations. The study of the interplay between the connections of system variables, linearization and multilevel solver for the overall complexity of the numerical method is of special interest. It is intended to develop and to use specially adapted nonlinear multilevel methods.

Examples Numerical solution of the Richards equation. Shown is a ca. 10m x 10m x 10m earth cube. It contains of four types of soil with different porosities and conductivities. Initial conditions a hydrostatic. On part of the surface there is constant precipitation. Shown are the different soils as well as transparent isosurfaces and a color encoded slice of the pressure over a period of 30 days (small movie, large movie). The computations have been done with PDELIB.

  • J. Fuhrmann, Zur Verwendung von Mehrgitterverfahren bei der numerischen Behandlung elliptischer partieller Differentialgleichungen mit variablen Koeffizienten, Shaker Verlag, Aachen, 1995.
  • T. Gerstner, F. Kiefer, A. Kunoth, Wavelet and multigrid methods for convection-diffusion equations, in "Dynamics of Multiscale Earth Systems", H.-J. Neugebauer (ed.), Lecture Notes in Earth Sciences. Springer, 2001. to appear.
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Thomas Gerstner