Natural Element Method And Its Application In Partial Differential Equation

Natural neighbor interpolation is a widely used multivariate scattered data interpolation method.Natural element method(NEM),which depends on the Voronoi diagram and Delaunay triangulation,is a numerical methods that natural neighbor interpolation(shape function)is used as trial and test approximations in a Galerkin method.Natural neighbor interpolation is only related to the distribution of nodes,nothing with the grid shape. Therefore, for the issue of irregular region, NEM has a natural advantage. Furthermore the shape function meets the interpolation property; and on the boundary of convex hull it is strictly linear continuous between two neighbor points, etc. so NEM not only has the advantages of the classical finite element(FEM) method and meshless method, but also overcomes some shortcomings of both.In this paper,we discuss NEM in which Sibson interpolation is used as shape function, and describe the properties,construction of shape function and the ap-plication of the NEM.First,we illustrate the definition of Sibson interpolation and give out some proofs involving smoothness,interpolation property, etc.Mean-while, we construct the key of the Galerkin method-shape function.Second, using a standard Galerkin method,we introduce the design and main structure of the NEM.Finally, we use NEM to solve the plane problem and hole plate prob-lem.In the experiment of hole plate problem,we compare the numerical results of the NEM and FEM.Since shape function is linear on the boundary of con-vex hull,we can conveniently impose the essential conditions on the boundary in the numerical experiment.At the same time,these examples also illustrate that NEM is very attractive,which is applied to solve partial differential equations of the irregular region.