  In this study we establish the existence and uniqueness of the solution of a
coupled system of general elliptic equations with anisotropic diffusion ,
non-uniform advection and variably influencing reaction terms on Lipschitz
continuous domain $\Omega \subset \mathbb{R}^m $ (m$\geq$1) with a Dirichlet
boundary. Later we consider the finite element (FE) approximation of the
coupled equations in a meshless framework based on weighted extended B-Spine
functions (WEBS).The a priori error estimates corresponding to the finite
element analysis are derived to establish the convergence of the corresponding
FE scheme and the numerical methodology has been tested on few examples.
