This work addresses the theoretical convergence and numerical solution of coupled partial differential equations (PDEs) with mixed dimensionality, specifically diffusion-advection equations posed in a three-dimensional domain coupled with a one-dimensional line. These systems arise in various scientific and engineering applications, where different physical phenomena are modeled in different spatial dimensions. The coupling between the three-dimensional and the one-dimensional PDEs relies on lateral averaging across an interface, facilitating the interaction between the 3D and 1D components. The primary focus is on the development and analysis of numerical schemes for solving such coupled systems, specifically using the finite element method (FEM) and the discontinuous Galerkin method. We consider two sets of boundary conditions. In the first problem, we impose the Neumann and Dirichlet boundary conditions at the boundary points of the one-dimensional PDE. Under suitable conditions on the magnitude of the advection field, we establish the existence and uniqueness of the weak solution, providing a theoretical foundation for the problem. We also formulate and analyze a finite element-based scheme, giving rigorous proofs of existence, uniqueness, and optimal priori error bound for the scheme. The results show that the scheme is theoretically sound and can be applied in practice. In the second problem, we consider another set of boundary conditions and assume that the coupled problem is well-posed. We develop a scheme that combines finite element method for the three-dimensional problem and the interior penalty discontinuous Galerkin method for the one-dimensional problem. We establish the existence and uniqueness of the discrete solution and also show the convergence of the method by deriving a priori error estimates.