On the Existence Uniqueness and Numerical Computation of Non-linear Coupled Elliptic PDE System with its application

TL;DR

This paper establishes the existence and uniqueness of solutions for a coupled non-linear elliptic PDE system, develops a finite element numerical scheme, proves its convergence, and applies it to free convection phenomena.

Contribution

It introduces a new finite element scheme for solving coupled non-linear elliptic PDEs with proven convergence and demonstrates its application to free convection problems.

Findings

Proved existence and uniqueness of solutions.

Developed a convergent finite element scheme.

Successfully applied the scheme to free convection phenomena.

Abstract

In this study we prove the existence-uniqueness of a coupled non-linear elliptic PDE system using Lax-Milgram theorem, Galerkin Method, Brouwer's fixed point theorem. Later we derive the finite element scheme for the numerical solution of the PDE system and also carry out the convergence analysis for the derived scheme. Further successfully apply the scheme to an application related to free convection phenomena.