Corner signularity for elliptic PDE

[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?

Thanks in advance!

 

[jvicens]

I would like to first commend you for your thorough analysis of the convergence rate of your spectral element method. This is an important step in understanding the behavior of your numerical solution and can lead to improvements in your method.

The weak singularity at the corners is a common issue in numerical methods, and it arises due to the discontinuity in the derivative of the solution at the corners. This discontinuity is not captured accurately by the spectral element method, which uses polynomial basis functions to approximate the solution.

To understand why the choice of the source function affects the convergence rate, we need to look at the behavior of the solution near the corners. For a simple case where f=1, the solution \psi will have a singularity at the corners, which can be approximated by a power law. The exponent of this power law depends on the choice of the source function f. If f is a smooth function, the exponent will be smaller, resulting in a weaker singularity and a faster convergence rate. However, if f has a singularity at the corners, the exponent will be larger, resulting in a stronger singularity and a slower convergence rate.

To analyze this problem in more detail, I recommend looking into the theory of singularities in partial differential equations. A good reference for this topic is the book "Singularities of Solutions of Second-Order Quasilinear Equations" by Vladimir Maz'ya.

In addition, there are some techniques that can be used to improve the convergence rate near the corners, such as using higher order basis functions or implementing special treatments for the corners. These techniques are discussed in the paper "Spectral element methods for singularly perturbed problems" by D. J. Knezevic and M. Ainsworth.

I hope this helps shed some light on the issue of weak singularities at the corners in your spectral element method. Keep up the good work in your research!