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In this study we prove the existence-uniqueness of a coupled non-linearelliptic PDE system using Lax-Milgram theorem, Galerkin Method, Brouwer's fixedpoint theorem. Later we derive the finite element scheme for the numericalsolution of the PDE system and also carry out the convergence analysis for thederived scheme. Further successfully apply the scheme to an application relatedto free convection phenomena.

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