On principles for the selection of shape functions for the Generalized Finite Element Method

Effective shape functions for the Generalized Finite Element Method should reflect the available information on the solution. This information is partially fuzzy, because the solution is, of course, unknown, and typically the only available information is the solution’s inclusion in a variety of function spaces. It is desirable to choose shape functions that perform robustly over a family of relevant situations. Quantitative notions of robustness are introduced and discussed. We show, in particular, that in one dimension polynomials are robust when the available information consists in inclusions in Sobolev-type spaces that are x-independent.

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