Institut für Computeranwendungen III, D-70569 Stuttgart

When doing adaptive simulations, in some cases optimal performance is only obtained when the algebraic data structure is closely connected with the geometry of the underlying grid. In this respect a pointer based matrix graph (such as implemented in the program package UG, see [1]) has several advantages over the classical Harwell-Boeing format. First, the graph structure allows local insertion/deletion of matrix elements in O(n) operations where n is the number of elements to be changed. This can be necessary if moving reaction zones are adaptively resolved. Second, reordering of the unknowns can be easily done, which is necessary to obtain robust multigrid smoothers in regions of dominating convection. Finally, the parallelization could be done on an abstract graph level.

Yet, for systems it is not reasonable to represent every unknown as a node and every matrix entry as a link in the graph, since this introduces a large overhead in both memory and computing time. On the other hand, storing full blocks on the links of the matrix graph is not adequate for large systems of reaction-convection-diffusion equations where the reaction terms usually couple only unknowns at one spatial location with each other. Therefore I introduce a new approach, where I allow the links in the matrix graph to represent sparse matrix blocks. Only the local sparsity pattern is given by an extended Harwell-Boeing type format permitting additionally identification of equal matrix elements.

In my talk, I will demonstrate the efficiency of this approach by several examples. These include 3D simulations of chemical reactions in stirred tanks (joint work with Sven Schmalzriedt, IBVT Stuttgart), as well as 3D simulations of biochemical reactions in saturated porous media (joint work with Christian Wagner, ICA3, Universität Stuttgart, and Rouslan Nabokov, IWR, Universität Heidelberg).

[1] P. Bastian, K. Birken, K. Johannsen, S. Lang, N. Neuss, H. Rentz-Reichert,
C. Wieners: *UG - A Flexible Software Toolbox for Solving Partial
Differential Equations.* Comput. Visual. Sci. 1, 27-40 (1997)