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The Babuška or plate paradox concerns the failure of convergence when a domain with curved boundary is approximated by polygonal domains in linear bending problems with simply supported boundary conditions. It can be explained via a boundary integral representation of the total Gaussian curvature that is part of the Kirchhoff–Love bending energy. It is shown that the paradox also occurs for a nonlinear bending-folding model which enforces vanishing Gaussian curvature. A simple remedy that is compatible with simplicial finite-element methods to avoid incorrect convergence is devised. 

Abstract We provide an a priori analysis of a certain class of numerical methods, commonly referred to as collocation methods, for solving elliptic boundary value problems. They begin with information in the form of point values of the right side $$f$$ of such equations and point values of the boundary function $$g$$ and utilize only this information to numerically approximate the solution $$u$$ of the partial differential equation (PDE). For such a method to provide an approximation to $$u$$ with guaranteed error bounds, additional assumptions on $$f$$ and $$g$$, called model class assumptions, are needed. We determine the best error (in the energy norm) of approximating $$u$$, in terms of the total number of point samples, under all Besov class model assumptions for the right-hand side and boundary data. We then turn to the study of numerical procedures and analyze whether a proposed numerical procedure (nearly) achieves the optimal recovery (OR) error. In particular, we analyze numerical methods that generate the numerical approximation to $$u$$ by minimizing specified data-driven loss functions over a set $$\varSigma $$ that is either a finite dimensional linear space, or more generally, a finite dimensional manifold. We show that the success of such a procedure depends critically on choosing a data-driven loss function that is consistent with the PDE and provides sharp error control. Based on this analysis, a loss function $${\cal L}^{*}$$ is proposed. We also address the recent methods of physics informed neural networks. We prove that minimization of the new loss $${\cal L}^{*}$$ over restricted neural network spaces $$\varSigma $$ provides an OR of the solution $$u$$, provided that the optimization problem can be numerically executed and $$\varSigma $$ has sufficient approximation capabilities. We also analyze variants of $${\cal L}^{*}$$ that are more practical for implementation. Finally, numerical examples illustrating the benefits of the proposed loss functions are given.