S-FDEM: The most flexible FDM for elliptic and parabolic PDEs

W. Schönauer, T. Adolph

Rechenzentrum der Universität Karlsruhe, D-76128 Karlsruhe, Germany

schoenauer@rz.uni-karlsruhe.de

Contributed talk

S-FDEM (Sliding Finite Difference Element Method) is a black-box solver for nonlinear systems of elliptic and parabolic PDEs on an arbitrary domain that may be composed of different subdomains with different PDEs and different unstructured 2-D or 3-D FEM grids. The domains may slide relatively to each other and the solution is coupled by coupling conditions (CCs). Individual mesh refinement in the subdomains is possible and there is a global error estimate for the whole solution.

The mathematical background for the FDM is elementary: local approximation of the solution by a polynomial of order $q$ in space (order $p$ in time), estimate of the discretization error by the order $q+2$ ($p+1$ in time). An error equation is derived whose terms are all computable which makes explicitly transparent the influence of all errors and thus opens the door for error balancing and selfadaptation. The development of a sophisticated algorithm with tuning parameters is the key to an efficient solution process (numerical engineering).

The FEM mesh (triangles in 2-D, tetrahedrons in 3-D) gives only the structure of the space. For the generation of difference and error formulas we search for adjacent nodes in rings/balls around the central node and select by a controlled pivoting the best nodes for high quality formulas. For practical reasons only the orders $q=2, 4, 6$ are relevant for technical applications. An optional comparison of the error terms for these orders permits the selection of an individual optimal order for each node (needs much storage for the formulas). The user prescribes a global relative tolerance tol and the mesh refinement process (by halving the edges of concerned elements) adapts in several cycles the solution to tol.

The coupling of different subdomains is by dividing lines (DLs) for FDEM or sliding dividing lines (SDLs) for S-FDEM. There the solution for the different subdomains (with different PDEs) is coupled by CCs. The SDLs allow non-matching grids and also the relative sliding of the subdomains. All resulting errors are included in the global error estimate . FDEM is implemented, S-FDEM is still under development.

The (iterative) solution of the resulting large and sparse linear systems of equations is by the LINSOL program package . It has presently 14 generalized CG methods and also polyalgorithms with automatic method switching. The matrix can be composed of 8 different basic data structures There is an (I)LU preconditioner and several bandwidth optimizers.

The whole software is efficiently parallelized with message passing on distributed memory parallel computers. Optimal data structures that support vector computers and microprocessors with cache are implemented.

In all these properties S-FDEM is a unique black-box solver for PDEs, including a global error estimate. Some examples illustrate the application of FDEM.