Operator-difference approximations on two-dimensional merged Voronoi-Delaunay grids

TL;DR

The paper addresses the challenge of solving boundary-value problems on irregular 2D domains by formulating them with coordinate-system invariant vector-analysis operators and implementing mimetic discretizations. It introduces a merged Voronoi-Delaunay grid (MVD) with orthodiagonal quadrilateral cells and separates scalar and vector quantities onto a main grid $\omega$ and an auxiliary grid $\omega^*$, enabling consistent operator-difference schemes for gradient $\operatorname{grad}_h$, divergence $\operatorname{div}_h$, and rotor $\operatorname{rot2D}_h$. These operators yield self-adjoint, positive-definite discrete systems for elliptic and rotor-type PDEs, such as $-\operatorname{div}_h(k \operatorname{grad}_h y) + c y = f$ and $\operatorname{rot2D}_h(k \operatorname{rot2D} y) + c y = f$, preserving discrete analogues of conservation and adjointness identities. The approach offers a robust, grid-geometry-flexible framework for 2D BVPs on complex domains, with direct applicability to scalar and vector problems and arbitrary irregular grids.

Abstract

Formulating boundary value problems for multidimensional partial derivative equations in terms of invariant operators of vector (tensor) analysis is convenient. Computational algorithms for approximate solutions are based on constructing grid analogs of vector analysis operators. This is most easily done by dividing the computational domain into rectangular cells when the grid nodes coincide with the cell vertices or are the cell centers. Grid operators of vector analysis for irregular regions are constructed using Delaunay triangulations or Voronoi partitions. This paper uses two-dimensional merged Voronoi-Delaunay grids to represent the grid cells as orthodiagonal quadrilaterals. Consistent approximations of the gradient, divergence, and rotor operators are proposed. On their basis, operator-difference approximations for typical stationary scalar and vector problems are constructed.

Figures (5)

• Figure 1: Nodes and grid cells in Delaunay triangulation and Voronoi partitioning
• Figure 2: MVD grid nodes and MVD grid cells in a two-dimensional domain
• Figure 3: MVD grid nodes and control volumes
• Figure 4: Gradient approximation on MVD grid
• Figure 5: Divergence approximation on the MVD grid