Discretized Volumes in Numerical Methods

TL;DR

This paper introduces two novel numerical methods involving discretized volumes and complex mappings to improve boundary value problem solutions, enabling symmetry analysis and solution transplantation between volumes.

Contribution

It presents new techniques for discretizing volumes and using complex mappings to transfer solutions, advancing numerical methods for boundary value problems.

Findings

Discretized volumes can be used to analyze symmetries in boundary value problems.

Complex mappings facilitate solution transfer between different volumes.

A correction function equation improves solution accuracy in volume transplantation.

Abstract

We present two techniques novel in numerical methods. The first technique compiles the domain of the numerical methods as a discretized volume. Congruent elements are glued together to compile the domain over which the solution of a boundary value problem is sought. We associate a group and a graph to that volume. When the group is symmetry of the boundary value problem under investigation, one can specify the structure of the solution, and find out if there are equispectral volumes of a given type. The second technique uses a complex mapping to transplant the solution from volume to volume and a correction function. Equation for the correction function is given. A simple example demonstrates the feasibility of the suggested method.