CVT-based Superconvergence And Its Application To Adaptive Finite Element Methods

The paper consists of two parts.In the first part,as we well know,usually it is impossible to generate an equilateral mesh for a general domain,even for a simple rectangular domain.Here,however,an almost equilateral triangular mesh for an arbitrary two dimensional bounded domain is generated by The Centroidal Voronoi Tesslation(CVT)-based methods.'Almost equilateral' means that almost all the triangular elements are close to being equilateral in terms of shape quality. Considering the Laplacian operator with Dirichlet Boundary condition,we find the linear finite element solutions have an O(h^2+α)(α≈0.5)-superconvergence at nodes on an almost equilateral triangular mesh generated based on CVT for an arbitrary two dimensional bounded domain.Extensive numerical examples are presented to demonstrate the superconvergence property.In the second part,motivated by the CVT-based mesh being nice property, we develop a convergent adaptive finite element method for elliptic problems in two dimensions based on CVT and superconvergent gradient recovery.Firstly, some kinds of gradient recovery methods are tested,which involve the weighted averaging,ZZ-SPR and PPR techniques.Based on the obtained error estimation, a new mesh sizing function is defined on each vertex by applying an error equal distribution principle.After re-meshing based on normalized edge lengths and optimization,the modified Lloyd iteration can be conducted for the construction of CVDT mesh to further improve and finally get a high quality mesh prepared for the new loop of adaption.Various numerical examples for elliptic equations are presented to demonstrate the effectiveness of the proposed approach.The innovations in this paper are as follows:We,for the first time,find the linear finite element solutions for the Laplacian operator with Dirichlet Boundary condition have an O(h^2+α)(α≈0.5)-superconvergence at nodes on an almost equilateral triangular mesh generated based on CVT for an arbitrary two dimensional bounded domain.Since all the existing implementations of the Lloyd iteration are done in a global manner,in the sense that the Delaunay triangulation of the newly computed mass centers are conducted globally which renders the whole Lloyd iteration procedure computation expensive.We presented a modified Lloyd iteration which is localized and accelerated in the following point replacement way.If the difference of the vertex and the newly computed mass center is small,the coordinates of the vertex are just replaced by those of the computed mass center;otherwise,local iteration is continued in order to keep the Delaunay property.The modified Lloyd iteration is significantly accelerated.In the adaptive FEM procedure,after each mesh Refine,due to the nice property of the CVT-based optimization,the updated triangular mesh is of high quality which guarantees the superconvergence property of the recovered gradients and the asymptotical exactness of the a posterior error estimate.Based on the equal distribution principle employed in the mesh sizing modification,the effectiveness of the error estimation leads to the convergence of the whole adaptation procedure.