Method of local element splittings for diffusion terms discretization in edge-bases schemes

Method of local element splittings is proposed for the discretization of the diffusion terms of the Navier – Stokes equations on mixed-element unstructured meshes. It is applicable when mesh functions are defined in nodes. This method is a linear method, which has much in common with the classical P1-Galerkin method. In the case of simplicial meshes, these methods coincide, but the new method yields a 7-point approximation of the 3D Laplace operator on a Cartesian mesh. On structured meshes the second order of accuracy is proved for the model heat equation. For general unstructured meshes, only the first order of accuracy is proved, however, the numerical evidence shows that there is no loss in accuracy in comparison with the classical P1-Galerkin method. The new method has an important advantage in the case of implicit time integration based on the Newton method, which implies solving linear algebraic systems with flux Jacobian. It allows to truncate flux Jacobian to a 7-point stencil for a much wider range of Reynolds and Courant numbers without loss of iterations convergence, compared to the P1-Galerkin method.