A Galerkin interface stabilisation method for the advection–diffusion and incompressible Navier–Stokes equations

A stabilised Galerkin finite element method for the advection–diffusion and incompressible Navier–Stokes equations is presented which inherits features of both discontinuous and continuous Galerkin methods. The problem of interest is posed element-wise with weakly imposed Dirichlet boundary conditions. The Dirichlet boundary condition is supplied by a function on element boundaries which is determined such that weak continuity of the flux is assured. The approach allows the natural incorporation of upwinding at element boundaries, which is typical of discontinuous Galerkin methods, while retaining the same number of global degrees of freedom as for a continuous Galerkin method. The formulation also stabilises mixed incompressible problems which use equal-order interpolations. For linear elements, only minor modifications are required to existing continuous finite element codes, and the link to other stabilised methods is elaborated. The method is supported by a range of numerical examples, which demonstrate stability with minimal numerical dissipation.