The locally conservative Galerkin (LCG) method for solving the incompressible Navier–Stokes equations

In this paper, the locally conservative Galerkin (LCG) method (Numer. Heat Transfer B Fundam. 2004; 46 :357–370; Int. J. Numer. Methods Eng. 2007) has been extended to solve the incompressible Navier–Stokes equations. A new correction term is also incorporated to make the formulation to give identical results to that of the continuous Galerkin (CG) method. In addition to ensuring element‐by‐element conservation, the method also allows solution of the governing equations over individual elements, independent of the neighbouring elements. This is achieved within the CG framework by breaking the domain into elemental sub‐domains. Although this allows discontinuous trial function field, we have carried out the formulation using the continuous trial function space as the basis. Thus, the changes in the existing CFD codes are kept to a minimum. The edge fluxes, establishing the continuity between neighbouring elements, are calculated via a post‐processing step during the time‐stepping operation. Therefore, the employed formulation needs to be carried out using either a time‐stepping or an equivalent iterative scheme that allows post‐processing of fluxes. The time‐stepping algorithm employed in this paper is based on the characteristic‐based split (CBS) scheme. Both steady‐ and unsteady‐state examples presented show that the element‐by‐element formulation employed is accurate and robust. Copyright © 2007 John Wiley & Sons, Ltd.