A modified finite element method for solving the time‐dependent, incompressible Navier‐Stokes equations. Part 1: Theory

Abstract Beginning with the Galerkin finite element method and the simplest appropriate isoparametric element for modelling the Navier‐Stokes equations, the spatial approximation is modified in two ways in the interest of cost‐effectiveness: the mass matrix is ‘lumped’ and all coefficient matrices are generated via 1‐point quadrature. After appending an hour‐glass correction term to the diffusion matrices, the modified semi‐discretized equations are integrated in time using the forward (explicit) Euler method in a special way to compensate for that portion of the time truncation error which is intolerable for advection‐dominated flows. The scheme is completed by the introduction of a subcycling strategy that permits less frequent updates of the pressure field with little loss of accuracy. These techniques are described and analysed in some detail, and in Part 2 (Applications), the resulting code is demonstrated on three sample problems: steady flow in a lid‐driven cavity at Re ≤ 10,000, flow past a circular cylinder at Re ≤ 400, and the simulation of a heavy gas release over complex topography.