<tt>pdelib</tt> - Algorithms and software components

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`pdelib` - Algorithms and software components for the numerical solution of partial differential equations

Collaborator: J. Fuhrmann, K. Gärtner, H. Langmach, M. Uhle, T. Streckenbach

Cooperation with: A. Linke, J. Bloch (Freie Universität Berlin (DFG Research Center MATHEON)), D. Hömberg (FG 4)

Supported by: DFG Research Center MATHEON, project C1, project C2

Description:

The purpose of this project is the further development of pdelib, a toolbox of software components for the numerical solution of partial differential equations. The re-design of the API and the code internals have reached the goal of first application projects.

Main features

Solver kernel Application Programming Interface (API) targeted at ease of use with OpenMP or pthreads on SMP computers. User callback routines operate on zones consisting of a certain number of elements rather than one element;
APIs for time step control, iterative solvers, preconditioners, bifurcation analysis;
fvsys API for implementing a solver for nonlinear systems of reaction-diffusion-convection equations using the finite volume method on simplicial grids;
Integration of the Delaunay mesh generators triangle [1] (2D) and TetGen (3D). This allows to describe geometries in the extension language Lua or in the C code and thus offers a large amount of flexibility for grid adaptation and geometry modification;
Grid partitioning for parallel computing and cache efficiency based on the METIS [4] code;
Parameter input and solver control can use the Lua [3] extension language;
Online visualization of the solution process based on OpenGL using the visualization tool gltools.
Components for medium complexity graphical user interfaces based on the FLTK GUI toolkit.

Focus during 2004

In close cooperation with project C1 ``Coupled systems of reaction-diffusion equations and application to the numerical solution of direct methanol fuel cell (DMFC) problems'' of the DFG Research Center MATHEON, we continued the development of tools for path following and bifurcation detection for systems of partial differential equations. The methods developed in this project have been successfully integrated with the pdelib2 solver kernel and the fvsys problem class. First benchmarks for two-dimensional problems have been run successfully.
The pdelib design was extended in order to allow the implementation of solvers using higher-order finite elements on simplices, focusing on second- and third-order elements. This new API in pdelib was applied and tested in the implementation of an experimental solver for the Oseen problem (see page ).
The build system was restructured using autoconf and automake. This was motivated by the fact that, despite of some drawbacks, this is the standard build system used in the Open Source Community. Based on this work, porting to MacOS X and Microsoft Windows is now possible.
Together with a general grid-to-grid interpolation code, Delaunay mesh generators have been tested successfully for implementing local adaptivity (see pages and ).

**Fig. 1:** Bifurcation analysis of a Brusselator model: one-dimensional domain (left) vs. two-dimensional domain (right)
$\makeatletter \@ZweiProjektbilderNocap[h]{0.47\textwidth}{fb04_fg3_pdelib_1}{fb04_fg3_pdelib_2} \makeatother$

References:

J.R. SHEWCHUK, Triangle: Engineering a 2D quality mesh generator and Delaunay triangulator, in: Applied Computational Geometry: Towards Geometric Engineering, M.C. Lin, D. Manocha, eds. Springer, Berlin, 1996, pp. 203-222.
B. SPITZAK ET AL., FLTK - the Fast Light Toolkit.
URL:http://www.fltk.org.
R. IERUSALIMSCHY, Programming in Lua, Lua.org, 2003, ISBN 85-903798-1-7.
URL: http://www.lua.org.
G. KARYPIS, V. KUMAR, METIS - Family of Multilevel Partitioning Algorithms.
URL: http://www-users.cs.umn.edu/ karypis/metis/