Matrix-Free Solution Algorithms in a Finite Element Context

摘要

An investigation was performed of the use of conjugate-gradient-type iterative solvers for the solution of fluid flow and heat transfer problems simulated with the finite element method. The literature indicates that the success of iterative methods for large sparse matrix equations depends critically on the choice of a suitable preconditioning scheme. Tests of several such schemes were performed on the discretized forms of the primitive governing equations (Navier-Stokes); it was found that the presence of the pressure term created difficulties in applying suitable preconditioners. Tests were also conducted on two other types of matrices: the symmetric, positive definite pressure matrix and the non-symmetric positive real advection-diffusion matrix. It was shown conclusively that these can be solved with appropriate preconditioners, with a dramatic reduction in storage requirements and CPU time for very large problems by comparison with direct solvers. The development of algorithms to reduce the fluid equations to these forms is discussed.