SJAM Winter-Spring 2002, Volume 3 - Number 1

Support operator method for Laplace equation on unstructured triangular grid

Victor Ganzha¹, Richard Liska², Mikhail Shashkov³ , Christoph Zenger¹

¹ Department of Informatics, Technical University of Munich, Arcisstrasse 21, 80333 Munich, Germany;
   email: ganzha@in.tum.de , zenger@in.tum.de

² Faculty of Nuclear Sciences and Physical Engineering, Czech Technical University in Prague, Břehová 7, 115 19 Prague 1, Czech Republic;
   email: liska@siduri.fjfi.cvut.cz

³ Group T-7, Los Alamos National Laboratory, Los Alamos, NM 87544, USA;
   email: misha@t7.lanl.gov

Received: February 27, 2002

Summary
A finite difference algorithm for solution of generalized Laplace equation on unstructured triangular grid is constructed by a support operator method. The support operator method first constructs discrete divergence operator from the divergence theorem and then constructs discrete gradient operator as the adjoint operator of the divergence. The adjointness of the operators is based on the continuum Green formulas which remain valid also for discrete operators. Developed method is exact for linear solution and has second order convergence rate. It is working well for  discontinuous diffusion coefficient and very rough or very distorted grids which appear quite often e.~g. in Lagrangian simulations. Being formulated on the unstructured grid the method can be used on the region of arbitrary geometry shape. Numerical results confirm these properties of the developed method.

Key words
mimetric finite difference, Laplace equation, unstructured  triangular grid
 

Mathematics Subject Classification (1991): 65N06, 65N12, 35J05

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