In the simulation of physical, biological and chemical processes one often has to deal with complex shaped domains. One might think of flow through root networks, solute transport on the pore scale of porous media or exchange processes through cell membranes.

We developed a new approach using a Discontinuous Galerkin discretization with trial and test functions defined on a structured grid [EB05]. This structured grid dictates the amount of degrees of freedom. We then trim the support of our trial and test functions according to the shape of the geometry.

We implemented the method within the DUNE [BDE+04] framework for 2D and 3D. Micro-scale computation are implemented, using domains which are implicitly given by a scalar function, i.e. CT image data or based on structural parameters generated artificial root structures (using the PlantVR software [CASJB06]). The iso-surface 0 gives the domain boundary. Local triangulations are obtained using Marching Cube/Simplex Algorithm (see [LC87]). In the domain we solve a potential equation. On the left and right boundary we impose dirichlet boundary conditions (left u=1, right u=0) and neumann boundary conditions everywhere else (grad u=0). We used test and trial functions of degree p=2.

[BDE+04]

P. Bastian, M. Droske, C. Engwer, R. Klöfkorn, T. Neubauer, M. Ohlberger, and M.Rumpf. Towards a unified framework for scientific computing. In Proceedings of the 15th Conference on Domain Decomposition Methods, LNCSE. Springer-Verlag, 2004.

[CASJB06]

S. Chuai-Aree, S. Siripant, W. Jäger, and H.G. Bock. PlantVR: Software for Simulation and Visualization of Plant Growth Model. In Proceedings of PMA06: The Second International Symposium on Plant growth Modeling, Simulation, Visualization and Applications. Beijing, 2006.

[EB05]

C. Engwer and P. Bastian. A Discontinuous Galerkin Method for Simulations in Complex Domains. Preprint (http://www.ub.uni-heidelberg.de/archiv/5707/), 2005.

[LC87]

William E. Lorensen and Harvey E. Cline. Marching cubes: A high resolution 3d surface construction algorithm. In SIGGRAPH '87: Proceedings of the 14th annual conference on Computer graphics and interactive techniques, pages 163-169, New York, NY, USA, 1987. ACM Press.