Adaptive Finite Element Methods for Elliptic PDEs Based on Conforming Centroidal Voronoi–Delaunay Triangulations

Abstract

Abstract.A new triangular mesh adaptivity algorithm for elliptic PDEs that combines a posteriori error estimation with centroidal Voronoi-Delaunay tessellations of domains in two dimensions is proposed and tested. The ability of the first ingredient to detect local regions of large error and the ability of the second ingredient to generate superior unstructured grids result in a mesh adaptivity algorithm that has several very desirable features, including the following. Errors are very well equidistributed over the triangles; at all levels of refinement, the triangles remain very well shaped, even if the grid size at any particular refinement level, when viewed globally, varies by several orders of magnitude; and the convergence rates achieved are the best obtainable using piecewise linear finite elements. This methodology can be easily extended to higher-order finite element approximations or mixed finite element formulations although only the linear approximation is considered in this paper.

Voronoi Tessellations and Their Application to Climate and Global Modeling

“…For one thing, through a point density function, CVT grid generation methodologies allow for a simple means of controlling the local grid size; moreover, the density function can easily be connected to error estimators, resulting in effective adaptive refinement strategies (Ju et al, 2002b). For another thing, CVTbased grids feature smooth transitions from coarse to fine grids; see Section 4.1.2 for an illustration.…”

“…The relation (9) between the density function and the local mesh sizes is also very useful in CVT-based adaptive mesh generation and optimization (Ju, 2007;Ju et al, 2002b).…”

Advances in Studies and Applications of Centroidal Voronoi Tessellations

“…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].…”

“…In [68,70], the concept of conforming CVT/CVDT was proposed with regard to these constraints which further elaborated the mixed energy minimization formulation given in [32]. Assume that the domain Ω is compact and ∂ Ω is piecewise smooth with singular/corner points…”

“…A special lifting process [68,70] is added during the Lloyd iteration that allows the boundary generators to return to the interior domain together with the projection process so that the number of boundary generators can vary freely according to the density function. Thus the dependence of resulting CfCVDT meshes on the initial guess is greatly weakened.…”