Study On Solution Methods For Sub-Systems Based On Preconditioned Navier-Stokes Equations

As for saddle point problems, it is very difficult to solve the equivalent linear sys-tems obtained from typical non-linear Navier-Stokes equations using linearization.Many scholars have proposed that using preconditioning technique changes the spectralproperties of the coefficient matrix of the equivalent linear system in order to be itera-tive method to achieve fast convergence. Since properties of ill-posed sparse matricesobtained from the saddle point problems after discretization are bad, they often need toconstruct efficient preconditioners to improve their spectral properties. Such as M. Ben-zi et al. have presented AL (Augmented Lagrangian) preconditioner and RDF (RelaxedDimensional Factorization) preconditioner, from their structure and actual use, theyshould be very quite competitive preconditioners.In this thesis, starting form practical implementation of the AL preconditioner andthe RDF preconditioner, we study the solution of a kind of subsystem (or it can be re-garded as the solving inverse problem of the matrix), thereby it can decrease theprocessing time for the throughout iterative process. For the sparse matrix generated onthe fine mesh, it’s not practical to directly solve inverse of submatrices in the process ofcomputing because size of matrices are large at this time. Therefore, it’s very necessaryto study how to efficiently solve these problems.At first, we introduce the AL preconditioner and the RDF preconditioner, thenfetch out our issue that studied. Since solving inverse of these type matrices is alwaysdealt with in the handling process of the AL and RDF preconditioners, and can be re-solved by direct LU factorization or iterative method, so we will conclude these twoprocessing strategies. According to the special structure of the submatrices, we combinewith this special sparse structure and present the processing method of the block LUfactorization and the block tridiagonal factorization, which is based on LU factorization;we also use the approximately solving inverse idea to analyze this issue. Thus they forma set of ideas to deal with such issue.Finally, we do numerical experiments for these methods proposed, and comparethem with the direct LU factorization. Experimental results show that these new strate- gies can significantly reduce the time-consuming of the entire process, especially blockdecomposition strategy. Because of the bad behavior of matrices, the approximate in-verse strategy may lead that the whole iterative process is stagnation and does not con-verge. For the problem on the fine grid, although the iterative strategy may reduce theconvergence rate of the entire solution process, the iterative processing method is moreeconomical than the direct method.