Harmonic Functions on Regions with Reentrant Corners. Part I.

It has been known for quite a while that if a function ux, y is harmonic in a region with reentrant corners, there are almost certainly infinite discontinuities of the first derivative of u in the neighborhood of the reentrant corner or corners. Simple examples are for an L-shaped region or T-shaped region. Some instances of these have been treated by conformally mapping the region into the interior of a rectangle. Attempts to solve the problem as first posed by a finite difference scheme or a finite element scheme will usually give poor approximations near any reentrant corner because the finite differences or finite elements have large truncation errors when a first derivative is infinite. When conformal mapping is tried, the conformal maps are usually only approximate, and similar errors arise, for more or less similar reasons. In view of recent work giving convergent expansions for u in the neighborhood of reentrant corners one can now give accurate solutions for such problems. Some experiments with such regions are reported.