Convergence improvement of the simultaneous relaxation method used in the finite element analysis of incompressible fluid flows

An improved finite element approach using the nine-node quadrilateral (Q2–Q1) elements is presented for the purpose of achieving faster convergence in simulations of incompressible fluid flows. The proposed numerical scheme employs a Gauss–Seidel-type or successive over-relaxation-type method in the numerical procedure based on the highly simplified marker-and-cell (HSMAC) method. Specifically, a key ingredient in the new numerical scheme is the incorporation of the other derivative terms in the first-order Taylor series expansion of the nodal-averaged error (in satisfying the equation of continuity) into the calculation for the simultaneous relaxation of velocity and pressure. The above-mentioned finite element approach is tested on classical fluid flow problems including the lid-driven square cavity flow and the flow past a circular cylinder. The results show significant improvement of the convergence property with the accuracy of the numerical solutions kept unchanged even on a highly distorted mesh.