Convergent Adaptive Finite Element Method Based on Centroidal Voronoi Tessellations and Superconvergence

Abstract

We present a novel adaptive finite element method (AFEM) for elliptic equations which is based upon the Centroidal Voronoi Tessellation (CVT) and superconvergent gradient recovery. The constructions of CVT and its dual Centroidal Voronoi Delaunay Triangulation (CVDT) are facilitated by a localized Lloyd iteration to produce almost equilateral two dimensional meshes. Working with finite element solutions on such high quality triangulations, superconvergent recovery methods become particularly effective so that asymptotically exact a posteriori error estimations can be obtained. Through a seamless integration of these techniques, a convergent adaptive procedure is developed. As demonstrated by the numerical examples, the new AFEM is capable of solving a variety of model problems and has great potential in practical applications.

Adaptive finite element analysis of elliptic problems based on bubble-type local mesh generation

“…In the first one, a new mesh is built according to a required mesh size. For example, the centroidal voronoi Delaunay triangulations have been developed for adaptive mesh generation and optimization by Ju [10] and Huang [12]. In the second family, a new mesh is obtained by dividing some selected elements.…”

Advances in Studies and Applications of Centroidal Voronoi Tessellations

“…This leads to another CVT/CVDT based adaptive finite element algorithm for numerical PDEs. For a number of model second-order elliptic problems with complex geometries and various singular solutions, the convergence of the adaptive algorithm was demonstrated in [67].…”

“…A numerical example from [70] is given in Figure 10. Superconvergence based a posterior error estimator was further developed in [67] for CVT/CVDT based meshes, utilizing their superconvergence properties as shown in [66]. This leads to another CVT/CVDT based adaptive finite element algorithm for numerical PDEs.…”

“…We remark that (4.1) is still a conjecture in spaces with d ≥ 2 but has been numerically verified by extensive experiments and is widely assumed in practical applications such as vector quantization and image processing [58]. The main idea of CVT/CVDT based adaptive algorithms is to first refine the old mesh and then optimize the mesh using the CVT/CVDT algorithms according to either the solution norms given in [32] or some other local a posteriori error estimators [67,70]. One can explicitly determine a density function ρ based on some a posteriori error estimator and the CVT mesh size relation (4.1) to generate the new mesh so that the errors of the new approximate solution will be equally distributed over the element in an optimal way [32,67,70].…”

“…The main idea of CVT/CVDT based adaptive algorithms is to first refine the old mesh and then optimize the mesh using the CVT/CVDT algorithms according to either the solution norms given in [32] or some other local a posteriori error estimators [67,70]. One can explicitly determine a density function ρ based on some a posteriori error estimator and the CVT mesh size relation (4.1) to generate the new mesh so that the errors of the new approximate solution will be equally distributed over the element in an optimal way [32,67,70]. In particular, an efficient nearest neighbor search algorithm proposed in [3] was used in [70] for the fast construction of the piecewise-smooth density function ρ over the triangulated domain.…”