# Corner signularity for elliptic PDE

1. Sep 14, 2013

### the_wolfman

I'm using spectral element methods to numerical solve a non-linear pde

$D \psi = f\left(x,\psi \right)$

in a rectangular domain, with $\psi = 0$ Here D is a second order elliptic operator.

I've found that the rate of convergence of my method depends on my choice of the functional form of my source $f\left(x,\psi \right)$. In some cases the error decreases geometrically as expected, but other choices result in reduced algebraic convergence. In the cases where algebraic convergence is observed, the error is dominated by the elements in the corners. This leads me to believe, that the reduced rate of convergence is due to the weak singularity at the corners. However, I'd still like to understand why for some choices of my source the weak singularity at the corners is important, and for other choices of my source it is not.

Qualitatively, I understand why a sharp corner is ill-behaved. But I really don't know where to being analysing
the problem.

Can anybody shed some light on how this singularity arises, even in the simple case where f=1. Or does anybody know a few good references that explain this problem in detail?