A strong form based moving Kriging collocation method for the numerical solution of partial differential equations with mixed boundary conditions

In this article, a novel mesh-free, moving Kriging (MK) based collocation scheme for the numerical solution of partial differential equations (PDEs) is introduced. In contrast to methods that are based on a Galerkin weak form of the governing PDEs, the MK collocation (MKC) approach, which is strong form based, is truly mesh-free in the sense that no background mesh is required for numerical integration. In fact, the presented approach does not require the evaluation of any integrals. Since the approximation function in the MK framework can be conditioned on point value- as well as derivative-information, the pointwise exact imposition of essential as well as natural boundary conditions is rendered straightforward. By incorporating an explicit linear basis into the MK framework, the first-order consistency condition is fulfilled, and thus rigid body motions are captured accurately. Moreover, Kriging functions may be conceived that comply with constraints on higher order derivatives such as the PDE at hand at certain locations. This possibility proves useful in improving the solution accuracy in the vicinity of Dirichlet boundaries. This article provides a study of the method's characteristics by means of 2D linear elasticity examples. It concludes with a suggestion on how to apply MKC to nonlinear PDEs.

Automated summary

This article introduces a novel mesh-free, moving Kriging based collocation scheme for numerical solution of partial differential equations (PDEs). The method is truly mesh-free, accurately imposes essential and natural boundary conditions, and improves solution accuracy.