Finite element analysis of boundary value problems on two-dimensional merged Voronoi-Delaunay grids
P.N. Vabishchevich1,2, M.M. Chernyshov1
1Lomonosov Moscow State University, Moscow, Russia
2Northeastern Federal University named after M.K. Ammosov, Yakutsk, Russia
Keywords: Delaunay triangulation, Voronoi partitioning, boundary value problem for second order elliptic equation, finite element method
Abstract
Delaunay triangulation and Voronoi partitioning are used to construct computational grids in numerical methods such as the finite element method and the finite volume method. A two-grid technique is considered that utilizes both the nodes of a Delaunay triangulation and the vertices of a Voronoi partitioning. This approach makes it possible to construct operator-difference approximations of the vector calculus operators (gradient, divergence, and curl) on a merged MVD (merged Voronoi-Delaunay) grid which consists of orthodiagonal quadrilaterals. The paper investigates an application of MVD grids for finite element analysis of two-dimensional boundary value problems using a Dirichlet problem for an elliptic equation in an anisotropic medium as an example. Two approaches are examined: Delaunay triangulation with additional Voronoi vertices as extra nodes and direct application of MVD grids. The results of computations on a sequence of progressively refined meshes employing different types of finite elements are presented.
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