?2011-4-20Efficient Preconditioners for SolvingSecond Order Mixed-type Finite VolumeElement Discretization SystemsCandidateXu SenlinSupervisorProfessor Shu ShiCollegeMathematics and Computational ScienceProgramThe Computational MathematicsSpecializationMultigrid Method and Domain Decomposition MethodDegreeMaster of ScienceUniversityXiangtan UniversityDate2011-4-20? ? ? ? ?(FVE)?.?,?,?.?,?,?- ILU?AMG?PGMRES(m)?.?,?,?.?,?.?,?:?.? ?,?,?.?,?,?PGMRES(m)?,?,?m?.? ?;?; PGMRES(m)?; ILU?; AMG?.IAbstractFinite Volume Element (FVE) method is one of the discretization methods for partialdifferential equations. Second order mixed-type finite volume element method which wasdeveloped recently has many advantages, such as being able to preserve local conservationof certain physical quantities. Fast algorithms for the corresponding discretization systemswill be discussed in this paper.Firstly, ILU- and AMG-GMRES(m) are employed to solve the linear systems arisingfrom second order mixed-type finite volume element scheme for an elliptic problem withjump coefficients. Numerical results show that the numbers of iteration of the two PGM-RES (m) are unstable which strongly depend on not only the mesh size but also the jumpcoefficient. Therefore, it is quite necessary to develop newly efficient preconditioners.Secondly,twonewpreconditionersaredesignedfortheabovelinearsystems,i.e., blockdiagonal and two-level preconditioner. For consistent mesh, theoretical analysis of the for-mer is given, and it is proved that the spectral condition number of the preconditioned sys-tem is uniformly bounded. Numerical results show that our preconditioners are stable andeffi cient. The numbers of iteration of the corresponding PGMRES (m) are significantly re-duced, furthermore, they are independent of mesh size and insensitiveto jump coefficient orrestart parameter.Key Words: finite volume element; preconditioner; PGMRES(m) method; ILU factoriza-tion; AMG method.II?.1?.32.1?.32.2?.4?PGMRES(m)?.83.1 PGMRES(m)?.83.2?.11?.154.1?.154.2?.21?.25?.26?.30?.31III?7, 11, 19, 24, 2729, 35,?(?)?,?.?(FVE)?.? ,?FVE?25, 26, 31, 40, 45, 48?,?FVE?.?,?FVE?8, 41, 49?.?8?,?,?,?,?.?,?,?,?.?,?,?GMRES(m)?4, 5, 39?,?,?,?.?,?.? ?:?LU (Incomplete LU Factorization, ILU)?1, 12, 15, 18, 22, 30, 33,?(Algebraic Multigrid, AMG)?2, 3, 6, 9, 10, 13, 14, 16, 20, 21, 32, 36, 37, 43, 44, 46, 47, 50,?,?.?, ILU?,?ILU(k)?ILUT?,?ILU(0)?,?. AMG?,?,?,?.?ILU(0)?AMG?PGMRES(m)?,?,?PGMRES(m)?,?,?,?.?,?,?m.?,?,?.1?,?A?,?.?,?,?A?Ad= diag(A11,A22)?,?A11?A22?.?A11,?,?AMG?,?AMG?A11?;?A22,?diag(A22)?,?,?