Finite Volume Difference Scheme for a Stiff Elliptic Reaction-Diffusion Problem with Line Interface
by
Ilia Braianov
Center of Applied Mathematics and Informatics, University of Rousse, 7017 Rousse, Bulgaria

Consider two dimensional elliptic problem in a rectangle, with Dirichle boundary conditions. Within the domain we suppose there is a regular interface (IN) across which the solution or some of its derivatives are known to be discontinuous and the source term can be also discontinuous or even singular. We deal with the case when the diffusion coefficients are small (multiplied by a small parameter) in the right part of the domain (RD). In this reason, boundary layers appear around the boundary of RD. The singularity of the domain and the discontinuity of the data cause also corner layers around corners of RD. It is well known that for problems with boundary and interior layers the standard numerical methods on uniform meshes are not uniformly convergent with respect to a small parameter. In order to solve numerically the problem we use finite volume method on a special condensed "Shishkin" mesh around the boundary of RD. The mesh that arise is not overlapping and we use interpolation to overcome this difficulty. Due to stiffness of the coefficients the matrix that we get is badly scaled but simple diagonal preconditioning improves the condition number. Uniformly convergence in a discrete energetic norm is proved. Numerical results confirming the theoretical estimates are given.

Date received: February 14, 2000