Equivalent operator preconditioning for linear elliptic problems

The numerical solution of linear elliptic partial differential equations most often involves a finite element or finite difference discretization. To preserve sparsity, the arising system is normally solved using an iterative solution method, commonly a preconditioned conjugate gradient method. Preconditioning is a crucial part of such a solution process. It is desirable that the total computational cost will be optimal, i.e. proportional to the degrees of freedom of the approximation used, which also includes mesh independent convergence of the iteration. This paper surveys the equivalent operator approach, which has proven to provide an efficient general framework to construct such preconditioners. Hereby one first approximates the given differential operator by some simpler differential operator, and then one chooses as preconditioner the discretization of this operator for the same mesh. In this survey we give a uniform presentation of this approach, including theoretical foundation and several practically important applications.

[1]  R. Temam.  Navier-Stokes Equations , 1977 .

[2]  Owe Axelsson,et al.  Symmetric Part Preconditioning of the CG Method for Stokes Type Saddle-Point Systems , 2007 .

[3]  Owe Axelsson,et al.  Mesh Independent Superlinear PCG Rates Via Compact-Equivalent Operators , 2007, SIAM J. Numer. Anal..

[4]  Louis A. Hageman,et al.  Iterative Solution of Large Linear Systems. , 1971 .

[5]  Y. Saad,et al.  GMRES: a generalized minimal residual algorithm for solving nonsymmetric linear systems , 1986 .

[6]  Owe Axelsson,et al.  On the updating and assembly of the Hessian matrix in finite element methods , 1988 .

[7]  Hervé Guillard,et al.  Iterative methods with spectral preconditioning for elliptic equations , 1990 .

[8]  P. G. Ciarlet,et al.  Three-dimensional elasticity , 1988 .

[9]  János Karátson,et al.  Superlinear PCG Algorithms: Symmetric Part Preconditioning and Boundary Conditions , 2008 .

[10]  G. Golub,et al.  Use of fast direct methods for the efficient numerical solution of nonseparable elliptic equations. , 1972 .

[11]  Tuomo Rossi,et al.  Parallel fictitious domain method for a non-linear elliptic neumann boundary value problem , 1999 .

[12]  James E. Gunn.  THE SOLUTION OF ELLIPTIC DIFFERENCE EQUATIONS BY SEMI-EXPLICIT ITERATIVE TECHNIQUES* , 1965 .

[13]  O. Axelsson,et al.  Algebraic multilevel preconditioning methods, II , 1990 .

[14]  P. Vassilevski,et al.  Algebraic multilevel preconditioning methods. I , 1989 .

[15]  Thomas A. Manteuffel,et al.  Preconditioning and boundary conditions without H 2 estimates: L 2 condition numbers and the distribution of the singular values , 1993 .

[16]  Solomon G. Mikhlin,et al.  The numerical performance of variational methods , 1971 .

[17]  James E. Gunn.  The numerical solution of ▿·a▿u=f by a semi-explicit alternating-direction iterative technique , 1964 .

[18]  Thomas A. Manteuffel,et al.  On the theory of equivalent operators and application to the numerical solution of uniformly elliptic partial differential equations , 1990 .

[19]  Gene H. Golub,et al.  Numerical solution of saddle point problems , 2005, Acta Numerica.

[20]  O. Axelsson.  Solving the Stokes problem on a massively parallel computer , 1999 .

[21]  Maya Neytcheva,et al.  Finite element block-factorized preconditioners , 2007 .

[22]  R. Bank.  Marching Algorithms for Elliptic Boundary Value Problems. II: The Variable Coefficient Case , 1977 .

[23]  Owe Axelsson,et al.  A black-box generalized conjugate gradient minimum residual method based on variable preconditioners and local element approximations , 2007 .

[24]  Owe Axelsson,et al.  Superlinearly convergent CG methods via equivalent preconditioning for nonsymmetric elliptic operators , 2004, Numerische Mathematik.

[25]  I. Hlavácek,et al.  Mathematical Theory of Elastic and Elasto Plastic Bodies: An Introduction , 1981 .

[26]  Graham F. Carey,et al.  Nonlinear preconditioned conjugate gradient and least-squares finite elements , 1987 .

[27]  Owe Axelsson,et al.  On the sublinear and superlinear rate of convergence of conjugate gradient methods , 2000, Numerical Algorithms.

[28]  O. Axelsson,et al.  Symmetric part preconditioning of the CGM for Stokes type saddle-point systems by , 2006 .

[29]  V. S. Vladimirov,et al.  Equations of mathematical physics , 1972 .

[30]  Solomon G. Mikhlin,et al.  Constants in Some Inequalities of Analysis , 1986 .

[31]  John William Neuberger,et al.  Sobolev gradients and differential equations , 1997 .

[32]  J. Karátson,et al.  Symmetric Part Preconditioning for the Conjugate Gradient Method in Hilbert Space , 2003 .

[33]  M. Schultz,et al.  Preconditioning by fast direct methods for nonself-adjoint nonseparable elliptic equations , 1986 .

[34]  Owe Axelsson,et al.  Sobolev space preconditioning for Newton's method using domain decomposition , 2002, Numer. Linear Algebra Appl..

[35]  Per Sundqvist,et al.  Numerical Computations with Fundamental Solutions , 2005 .

[36]  P. Ciarlet,et al.  Mathematical elasticity, volume I: Three-dimensional elasticity , 1989 .

[37]  T. Manteuffel,et al.  A taxonomy for conjugate gradient methods , 1990 .

[38]  Yousef Saad,et al.  Iterative methods for sparse linear systems , 2003 .

[39]  Tuomo Rossi,et al.  A Parallel Fast Direct Solver for Block Tridiagonal Systems with Separable Matrices of Arbitrary Dimension , 1999, SIAM J. Sci. Comput..

[40]  Tamás Kurics,et al.  On Superlinear PCG Methods for FDM Discretizations of Convection-Diffusion Equations , 2008, NAA.

[41]  Tamás Kurics,et al.  Superlinearly convergent PCG algorithms for some nonsymmetric elliptic systems , 2008 .

[42]  Thomas A. Manteuffel,et al.  Preconditioning Second-Order Elliptic Operators: Experiment and Theory , 1992, SIAM J. Sci. Comput..

[43]  曹志浩,et al.  ON ALGEBRAIC MULTILEVEL PRECONDITIONING METHODS , 1993 .

[44]  O. Widlund.  A Lanczos Method for a Class of Nonsymmetric Systems of Linear Equations , 1978 .

[45]  Ian G. Graham,et al.  Unstructured Additive Schwarz-Conjugate Gradient Method for Elliptic Problems with Highly Discontinuous Coefficients , 1999, SIAM J. Sci. Comput..

[46]  Owe Axelsson,et al.  Conditioning analysis of separate displacement preconditioners for some nonlinear elasticity systems , 2004, Math. Comput. Simul..

[47]  István Faragó,et al.  Numerical solution of nonlinear elliptic problems via preconditioning operators : theory and applications , 2002 .

[48]  Olof B. Widlund,et al.  On the use of Fast Methods for Separable Finite Difference Equations for the Solution of General Elliptic Problems , 1972 .

[49]  Anne Greenbaum,et al.  Fast Parallel Iterative Solution of Poisson's and the Biharmonic Equations on Irregular Regions , 2011, SIAM J. Sci. Comput..

[50]  Johannes K. Kraus,et al.  Algebraic multilevel preconditioning of finite element matrices using local Schur complements , 2006, Numer. Linear Algebra Appl..

[51]  O. Widlund,et al.  On finite element domain imbedding methods , 1990 .

[52]  R. Winther.  Some Superlinear Convergence Results for the Conjugate Gradient Method , 1980 .

[53]  Sergey Repin,et al.  A posteriori error estimation for the Poisson equation with mixed Dirichlet/Neumann boundary conditions , 2004 .

[54]  Joseph E. Pasciak,et al.  Preconditioned Iterative Methods for Nonselfadjoint or Indefinite Elliptic Boundary Value Problems. , 1984 .

[55]  Anne Greenbaum.  Diagonal Scalings of the Laplacian as Preconditioners for Other Elliptic Differential Operators , 1992, SIAM J. Matrix Anal. Appl..

[56]  R. Temam,et al.  Navier-Stokes equations: theory and numerical analysis: R. Teman North-Holland, Amsterdam and New York. 1977. 454 pp. US $45.00 , 1978 .

[57]  Owe Axelsson,et al.  Variable-step multilevel preconditioning methods, I: Self-adjoint and positive definite elliptic problems , 1994, Numer. Linear Algebra Appl..

[58]  Ronald D. Haynes,et al.  Preconditioning for a Class of Spectral Differentiation Matrices , 2005, J. Sci. Comput..

[59]  István Faragó,et al.  Variable Preconditioning via Quasi-Newton Methods for Nonlinear Problems in Hilbert Space , 2003, SIAM J. Numer. Anal..

[60]  Radim Blaheta,et al.  Displacement decomposition - incomplete factorization preconditioning techniques for linear elasticity problems , 1994, Numer. Linear Algebra Appl..

[61]  Anne Greenbaum,et al.  Iterative methods for solving linear systems , 1997, Frontiers in applied mathematics.

[62]  Zahari Zlatev,et al.  Computer Treatment of Large Air Pollution Models , 1995 .

[63]  Gene H. Golub,et al.  Inexact Preconditioned Conjugate Gradient Method with Inner-Outer Iteration , 1999, SIAM J. Sci. Comput..

[64]  Owe Axelsson,et al.  On iterative solvers in structural mechanics; separate displacement orderings and mixed variable methods , 1999 .

[65]  O. Axelsson.  A generalized conjugate gradient, least square method , 1987 .

[66]  D. Funaro.  Polynomial Approximation of Differential Equations , 1992 .

[67]  Barry Lee,et al.  Finite elements and fast iterative solvers: with applications in incompressible fluid dynamics , 2006, Math. Comput..

[68]  István Antal,et al.  A mesh independent superlinear algorithm for some nonlinear nonsymmetric elliptic systems , 2008, Comput. Math. Appl..

[69]  Owe Axelsson,et al.  AN ITERATIVE SOLUTION METHOD FOR SCHUR COMPLEMENT SYSTEMS WITH INEXACT INNER SOLVER , 1999 .

[70]  P. Swarztrauber.  A direct Method for the Discrete Solution of Separable Elliptic Equations , 1974 .

[71]  Owe Axelsson,et al.  A combined method of local Green's functions and central difference method for singularly perturbed convection-diffusion problems , 2003 .

[72]  János Karátson.  On the Superlinear Convergence Rate of the Preconditioned CGM for Some Nonsymmetric Elliptic Problems , 2007 .

[73]  I. Gustafsson.  A class of first order factorization methods , 1978 .

[74]  M. Hestenes,et al.  Methods of conjugate gradients for solving linear systems , 1952 .

[75]  S. Eisenstat,et al.  Variational Iterative Methods for Nonsymmetric Systems of Linear Equations , 1983 .

[76]  Andrew V. Knyazev,et al.  Steepest Descent and Conjugate Gradient Methods with Variable Preconditioning , 2007, SIAM J. Matrix Anal. Appl..

[77]  T. Manteuffel.  The Tchebychev iteration for nonsymmetric linear systems , 1977 .

[78]  Ivan Lirkov,et al.  A Parallel Algorithm for Systems of Convection-Diffusion Equations , 2006, Numerical Methods and Applications.

[79]  Johannes C. C. Nitsche,et al.  Error estimates for the numerical solution of elliptic differential equations , 1960 .

[80]  Ye.G. D'yakonov.  The construction of iterative methods based on the use of spectrally equivalent operators , 1966 .

[81]  Valeria Simoncini,et al.  Flexible Inner-Outer Krylov Subspace Methods , 2002, SIAM J. Numer. Anal..

[82]  Owe Axelsson,et al.  Scalable algorithms for the solution of Navier's equations of elasticity , 1995 .

[83]  Owe Axelsson,et al.  Preconditioning of Boundary Value Problems Using Elementwise Schur Complements , 2009, SIAM J. Matrix Anal. Appl..

[84]  O. Nevanlinna.  Convergence of Iterations for Linear Equations , 1993 .

[85]  Thomas A. Manteuffel,et al.  Optimal equivalent preconditioners , 1993 .

[86]  T. Manteuffel,et al.  Necessary and Sufficient Conditions for the Existence of a Conjugate Gradient Method , 1984 .

[87]  Cornelius O. Horgan,et al.  Korn's Inequalities and Their Applications in Continuum Mechanics , 1995, SIAM Rev..

[88]  P. G. Ciarlet,et al.  Maximum principle and uniform convergence for the finite element method , 1973 .

[89]  Philippe G. Ciarlet,et al.  The finite element method for elliptic problems , 2002, Classics in applied mathematics.

[90]  Owe Axelsson,et al.  Diagonally compensated reduction and related preconditioning methods , 1994, Numer. Linear Algebra Appl..

[91]  János Karátson,et al.  Mesh independent superlinear convergence estimates of the conjugate gradient method for some equivalent self-adjoint operators , 2005 .

[92]  P. Swarztrauber.  THE METHODS OF CYCLIC REDUCTION, FOURIER ANALYSIS AND THE FACR ALGORITHM FOR THE DISCRETE SOLUTION OF POISSON'S EQUATION ON A RECTANGLE* , 1977 .