On the control-volume finite-element method for convection–diffusion equations on nonconforming triangular meshes

We examine the formulation of the control-volume finite-element method (CVFEM) for convection–diffusion equations on nonconforming triangular meshes. We present an approach for treating nonconforming element interfaces within the CVFEM framework and verify it through a series of numerical experiments. In addition, we explore the relationship between the CVFEM and the Galerkin finite-element method (FEM) with linear elements. We show that the element-level formulations of the two methods can be cast in an algebraically equivalent form when a constant-flux approximation is applied within the element interior. This equivalence provides a locally conservative interpretation of the Galerkin FEM. Moreover, the CVFEM offers a simple geometric interpretation of both artificial-diffusion stabilization terms and mass-matrix lumping commonly used in finite-element methods. The global formulations of the CVFEM and the Galerkin FEM can be made algebraically equivalent on conforming triangular meshes. On nonconforming triangular meshes, however, the methods differ in their treatment of contributions associated with hanging nodes. In contrast to the Galerkin FEM, the CVFEM provides a locally conservative formulation in the vicinity of nonconforming element interfaces. Numerical experiments show that both methods achieve comparable accuracy. Nevertheless, the CVFEM may be preferable because of its locally conservative formulation. This property can be useful when the CVFEM is combined with adaptive mesh refinement techniques, as it provides a clear definition of the discrete quantities that must be conserved during mesh refinement and coarsening operations. • Control-volume finite-element method (CVFEM) on nonconforming triangular meshes. • Relationship between the CVFEM and the Galerkin method. • Geometric interpretation of stabilization terms and mass-matrix lumping.