VAGO method for the solution of elliptic second-order boundary value problems

TL;DR

The paper introduces the VAGO method, which uses vector analysis operators constructed from Delaunay and Voronoi diagrams to solve elliptic second-order boundary value problems approximately.

Contribution

It presents a novel operator-based discretization approach for elliptic PDEs using Delaunay and Voronoi structures, bridging differential operators with grid-based methods.

Findings

Effective approximation of elliptic boundary value problems.

Application to convection-diffusion-reaction equations with tensor diffusion.

Utilization of Delaunay and Voronoi diagrams for operator construction.

Abstract

Mathematical physics problems are often formulated using differential oprators of vector analysis - invariant operators of first order, namely, divergence, gradient and rotor operators. In approximate solution of such problems it is natural to employ similar operator formulations for grid problems, too. The VAGO (Vector Analysis Grid Operators) method is based on such a methodology. In this paper the vector analysis difference operators are constructed using the Delaunay triangulation and the Voronoi diagrams. Further the VAGO method is used to solve approximately boundary value problems for the general elliptic equation of second order. In the convection-diffusion-reaction equation the diffusion coefficient is a symmetric tensor of second order.