Conjugate gradient methods and ILU preconditioning of non‐symmetric matrix systems with arbitrary sparsity patterns

Abstract

AbstractPreconditioning techniques based on incomplete Gaussian elimination for large, sparse, non‐symmetric matrix systems are described. A certain level of fill‐in may be specified in the incomplete factorizations. All methods considered may be applied to matrices with arbitrary sparsity patterns, for instance those associated with the general preprocessor algorithms or adaptive mesh techniques. The preconditioners have been combined with five conjugate gradient‐like methods and tested on finite element discretized scalar convection‐diffusion equations in 2D and 3D. It is found from numerical experiments that an amount of fill‐in corresponding to about 50% of the number of original non‐zero matrix entries is the optimal choice for this class of preconditioners. The preconditioners show almost no sensitivity to grid distortion. In problems with significantly variable coefficients or anisotropy the preconditioners stabilize the basic iterative schemes in addition to reducing the computational work substantially, mostly by more than 90%. The modified preconditioning technique, where fill‐in is added on the main diagonal, performs in general better than the standard incomplete LU factorization, but is inferior to the latter in 3D problems and for matrix systems with complicated sparsity patterns.