Dealing with singularities: approximation with enriched bases and function spaces

Very often, one has more knowledge about the solution to a computational problem than can be matched to a known basis. For example, a function may have singularities at known points, such as at corners of a domain, or it may have certain oscillatory behavior or even combinations of those features. However, augmenting or enriching a basis to incorporate that behavior typically makes it redundant: it ceases to be a basis. Correspondingly, numerical discretization of the approximation problem tends to produce ill-conditioned matrices. Is that necessarily a problem? We explore some ground rules of approximation with enriched bases, we provide a mathematical analysis and complement this with efficient numerical algorithms. The combined insights enable a multitude of novel approaches to deal with singularities of functions.