Preconditioning in Hdiv and applications

We consider the solution of the system of linear algebraic equations which arises from the finite element discretization of boundary value problems associated to the differential operator I-grad div. The natural setting for such problems is in the Hilbert space H(div) and the variational formulation is based on the inner product in H(div). We show how to construct preconditioners for these equations using both domain decomposition and multigrid techniques. These preconditioners are shown to be spectrally equivalent to the inverse of the operator. As a consequence, they may be used to precondition iterative methods so that any given error reduction may be achieved in a finite number of iterations, with the number independent of the mesh discretization. We describe applications of these results to the efficient solution of mixed and least squares finite element approximations of elliptic boundary value problems.

[1]  T. Manteuffel,et al.  First-order system least squares for second-order partial differential equations: part I , 1994 .

[2]  J. Pasciak,et al.  Convergence estimates for product iterative methods with applications to domain decomposition , 1991 .

[3]  Zhangxin Chen,et al.  Domain decomposition algorithms for mixed methods for second-order elliptic problems , 1996, Math. Comput..

[4]  Tarek P. Mathew,et al.  Schwarz alternating and iterative refinement methods for mixed formulations of elliptic problems, part I: Algorithms and numerical results , 1993 .

[5]  Panayot S. Vassilevski,et al.  Interior penalty preconditioners for mixed finite element approximations of elliptic problems , 1996, Math. Comput..

[6]  W. Hackbusch,et al.  A New Convergence Proof for the Multigrid Method Including the V-Cycle , 1983 .

[7]  M. Fortin,et al.  ERROR ANALYSIS OF MIXED-INTERPOLATED ELEMENTS FOR REISSNER-MINDLIN PLATES , 1991 .

[8]  Jinchao Xu,et al.  Iterative Methods by Space Decomposition and Subspace Correction , 1992, SIAM Rev..

[9]  R. S. Falk,et al.  Error estimates for mixed methods , 1980 .

[10]  Apostol T. Vassilev,et al.  Analysis of the Inexact Uzawa Algorithm for Saddle Point Problems , 1997 .

[11]  P. Vassilevski,et al.  Multilevel iterative methods for mixed finite element discretizations of elliptic problems , 1992 .

[12]  J. Pasciak,et al.  A preconditioning technique for indefinite systems resulting from mixed approximations of elliptic problems , 1988 .

[13]  D. Arnold,et al.  Preconditioning discrete approximations of the Reissner-Mindlin plate model , 1997 .

[14]  G. Carey,et al.  Least-squares mixed finite elements for second-order elliptic problems , 1994 .

[15]  Charles I. Goldstein,et al.  Multilevel Iteration for Mixed Finite Element Systems with Penalty , 1993, SIAM J. Sci. Comput..

[16]  G. Golub,et al.  Inexact and preconditioned Uzawa algorithms for saddle point problems , 1994 .

[17]  Susanne C. Brenner.  A multigrid algorithm for the lowest-order Raviart-Thomas mixed triangular finite element method , 1992 .

[18]  Ping Lin.  A Sequential Regularization Method for Time-Dependent Incompressible Navier--Stokes Equations , 1997 .

[19]  Panayot S. Vassilevski,et al.  Domain Embedding Preconditioners For Mixed Systems , 1998 .

[20]  Panayot S. Vassilevski,et al.  An application of the abstract multilevel theory to nonconforming finite element methods , 1995 .

[21]  Panayot S. Vassilevski,et al.  Preconditioning Mixed Finite Element Saddle‐point Elliptic Problems , 1996 .