A finite element solver for the Navier-Stokes equations using a preconditioned adaptive BICGSTAB (l) method.

摘要

The physical behavior of many fluids is governed by the incompressible Navier-Stokes equations. The equations are far too complex to be computed analytically and hence numerical solutions are necessary. To study the numerical solution of these equations, a computer program, called BCSLNSS, was written in Fortran-90. Fortran-90 is an extension of Fortran-77 that includes many of the features of C. The most valuable of these is the dynamic memory allocation of vectors and arrays. BCSLNSS solves the 2-dimensional, incompressible Navier-Stokes equations in a primitive variable formulation on polygonal Dirichlet domains. Discretizations in space and time are performed using the P1-Bubble-P1 finite element method and a one-step {dollar}theta{dollar} method respectively. Krylov subspace methods are used to solve the resulting linear systems. The conjugate gradient method generally will not perform well because the linear systems are nonsymmetric. This leads to the class of BICG methods. A recent member in this class is the BICGSTAB(l) algorithm. In this paper, a method known as ASTAB(l) is proposed. This algorithm takes advantage of the fact that the value of l can be adaptively selected. Experiments were performed with BCSLNSS to assess the validy of the code. In addition, the ASTAB(l) algorithm was compared to the BICGSTAB(l) algorithm to see if any advantage is gained in the adaptive strategy. The results indicate that the adaptive method is more robust, but generally not more efficient. Finally, experiments were performed to determine suitable preconditioners when solving the velocity and pressure equations. It was found that the ILUT(m) preconditioner is outstanding in comparison to diagonal preconditioners for convection-diffusion solves. Selecting a suitable pressure preconditioner is a difficult task because the coefficient matrix is never available. In spite of this the application of the ILUT(m) preconditioner to simple approximations of the pressure matrix seems promising.