I was recently attended some talks on numerical approaches to integral equations. From what I gather, if an operator in the integral equation is not compact it leads to problems numerically. An example being in the case of a domain with a corner - this leads to an integral equation with a non-compact operator and is supposedly a tougher problem to handle than a domain with a smooth boundary.
One thing I didn't pick up at the talks, is why specifically are problems with non-compact operators tougher to handle numerically? I believe it has something to do with the spectrum not being discrete in the case of non-compactness but I'd really like some clarification on the precise difficulties of handling non-compact operators numerically?