Quadrature for meshless Nitsche's method (Q2874177)

The author presents a study on the effect of numerical integration on a Galerkin meshless method (GMM), applied to approximate solutions of elliptic partial differential equations with essential boundary conditions (EBC). It is well known that it is difficult to impose the EBC on the standard approximation space used in GMM. The author uses Nitsche's approach, which was introduced in the context of the finite element method, to impose the EBC. It is required that the numerical integration rule satisfies (a) a `discrete Green's identity' on polynomial spaces, and (b) a `conforming condition' involving the additional integration terms introduced by Nitsche's approach. Based on such numerical integration rules, the author obtains a convergence result for the meshless Nitsche method (MNM) with numerical integration, where the shape functions reproduce polynomials of degree \(k\geq\,1\). The analysis could be extended to the symmetric MNM. Numerical results are presented to illuminate the theoretical results and to demonstrate the efficiency of the algorithms.