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

TL;DR

This paper develops operator-difference approximation methods on merged Voronoi-Delaunay grids for solving boundary value problems in multidimensional partial differential equations, focusing on irregular regions.

Contribution

It introduces new consistent operator-difference approximations on merged Voronoi-Delaunay grids for vector analysis operators in irregular domains.

Findings

Effective approximation of gradient, divergence, and rotor operators.

Construction of operator-difference schemes for scalar and vector problems.

Application to stationary boundary value problems.

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.