Convergence of the Finite Difference Scheme for a Fast Diffusion Equation in Porous Media

Abstract

We are concerned with an implicit scheme for the finite difference solution to a nonlinear parabolic equation with a multivalued coefficient that describes the fast diffusion in a porous medium. The boundary conditions contain the multivalued function as well. We prove the stability and the convergence of the scheme, emphasizing the precise nature of convergence in this specific case, and compute the error level of the approximating solution. The method is aimed to simplify the numerical computations for the solutions to equations of this type, without performing an approximation of the multivalued function. The theory is illustrated by numerical results.

Keywords:

• Boundary value problems for nonlinear parabolic PDE
• Flows in porous media
• Free boundary problems for PDE
• PDE with multivalued right-hand sides
• Stability and convergence of numerical methods

AMS Subject Classification:

• 35K60
• 35R70
• 35R35
• 65M12
• 76S05

ACKNOWLEDGMENT

This work was carried out as part of the projects PN II IDEI ID_404 and CEEX-05-D11-36/2005, financed by the Romanian Ministry of Education, Research and Youth.