Grid Generation And Application Of The Delaunay Tringulation Under A New Metric

The unstructured mesh has been widely applied to the numerical calculation procedure of science and engineering.Delaunay triangularization is one of the important methods of generating the unstructured meshes.The characteristics of Delaunay triangular mesh are as follows,firstly,the generated triangles never overlap each other,secondly,all of the generated triangles cover the whole region, thirdly,the circumcircle of the generated triangle has never include other points. Many scholars have done a lot of work on mesh generation and applications based on Delaunay triangular mesh,but transforming the distance metric,mesh generation based on the Delaunay triangulation method is relatively small.We apply the meshes to solve PDE,For linear finite elements on triangular meshes,the optimal global convergence rate of the function values in the L₂ norm is O(h²),where h is the mesh size,and this result cannot be improved in general.For the Poisson equation with Dirichlet boundary conditions,if the mesh consists of equilateral triangles,the error of the function values is of order O(h⁴) at nodes,which is two orders higher than the optimal global convergent rate,it is proved by Blum,Lin, Rannacher.If we choose the appropriate mesh,we further analysis super convergence for elliptic partial differential equations of second order.In this paper,the primary pursuits are as follows.Firstly,the distance metric is a crucial part of the unstructured mesh generation by Delaunay triangulation method.In the process of Delannay triangulation, transforming to another distance metric,we generate Delaunay triangle mesh under the new metric,and apply the mesh to solve the elliptic partial differential equations of second order.Secondly,for the elliptic partial differential equations with const coefficients, we find the linear transformation matrix T.Equilateral triangleτis changed to another triangleτ′by inverse transformation T^-1,we analysis convergence for the elhptic partial differential equations of second order by unit analysis method,and get super convergence property of the nodal function values occurs on the mesh consists of triangleτ′,the super convergence order is O(h⁴).Numerical results confirming the theory are presented.