The variational collocation method

We propose the variational collocation method for the numerical solution of partial differential equations. The conceptual basis is the establishment of a direct connection between the Galerkin method and the classical collocation methods, with the perspective of achieving the accuracy of the former with a computational cost of one point evaluation per degree of freedom as in the latter. Variational collocation requires a discrete space constructed by smooth and pointwise non-negative basis functions, which makes the approach immediately applicable to isogeometric analysis and some meshfree methods. In this paper, we concentrate on isogeometric analysis and demonstrate that there exists a set of points such that collocation of the strong form at these points produces the Galerkin solution exactly. We provide an estimate of these points and show that applying isogeometric collocation at the estimated points completely solves the well-known odd/even discrepancy in the order of spatial convergence. We demonstrate the potential of variational collocation with examples of linear and non-linear elasticity as well as Kirchhoff plates.