Finite Difference Approximation at Re-entrant Corners.
by
David Jarvis
Dept. of Anaesthesia, Royal Adelaide Hospital, University of Adelaide.
Coauthors: John Noye (Dept. of Applied Mathematics, University of Adelaide)

The numerical approximation to solutions of parabolic or elliptic partial differential equations by finite difference methods may be inaccurate at points of singularity on the boundary, such as a re-entrant corner, when unmodified finite difference formulations are used. A simple modification of the finite difference approximation to the solution at the point of singularity on a re-entrant corner of internal angle 270 degrees, based on the continuity of the solution but not on its derivatives, restores the accuracy of the scheme without adding to the complexity of the solution algorithm. The numerical solution to a model time dependent diffusion equation in a bounded region containing singularities due to the presence of re-entrant corners is given. The results are compared to those of the unmodified finite difference scheme and those refined by local approximation to the analytical solution in the region of the singularity. Furthermore, convergence can be improved using standard finite difference techniques such as extrapolation.

Date received: July 30, 1999