Hierarchical bases and the finite element method

In this work we present a brief introduction to hierarchical bases, and the important part they play in contemporary finite element calculations. In particular, we examine their role in a posteriori error estimation, and in the formulation of iterative methods for solving the large sparse sets of linear equations arising from finite element discretization.

[1]  J. Pasciak,et al.  Parallel multilevel preconditioners , 1990 .

[2]  H. Yserentant.  Old and new convergence proofs for multigrid methods , 1993, Acta Numerica.

[3]  W. Rheinboldt,et al.  Error Estimates for Adaptive Finite Element Computations , 1978 .

[4]  A. Aziz.  The Mathematical Foundations of the Finite Element Method with Applications to Partial Differential Equations , 1972 .

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

[6]  Michael Griebel,et al.  Multilevel Algorithms Considered as Iterative Methods on Semidefinite Systems , 1994, SIAM J. Sci. Comput..

[7]  Randolph E. Bank,et al.  PLTMG - a software package for solving elliptic partial differential equations: users' guide 8.0 , 1998, Software, environments, tools.

[8]  J. Oden,et al.  A unified approach to a posteriori error estimation using element residual methods , 1993 .

[9]  Ivo Babuška,et al.  Accuracy estimates and adaptive refinements in finite element computations , 1986 .

[10]  Ivo Babuška,et al.  Basic principles of feedback and adaptive approaches in the finite element method , 1986 .

[11]  J. Oden,et al.  A procedure for a posteriori error estimation for h-p finite element methods , 1992 .

[12]  Victor Eijkhout,et al.  The Role of the Strengthened Cauchy-Buniakowskii-Schwarz Inequality in Multilevel Methods , 1991, SIAM Rev..

[13]  Gene H. Golub,et al.  Matrix computations , 1983 .

[14]  L. R. Scott,et al.  On the conditioning of finite element equations with highly refined meshes , 1989 .

[15]  L. R. Scott,et al.  The Mathematical Theory of Finite Element Methods , 1994 .

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

[17]  Gene H. Golub,et al.  Some History of the Conjugate Gradient and Lanczos Algorithms: 1948-1976 , 1989, SIAM Rev..

[18]  Ricardo G. Durán,et al.  On the asymptotic exactness of Bank-Weiser's estimator , 1992 .

[19]  R. P. Kendall,et al.  An Approximate Factorization Procedure for Solving Self-Adjoint Elliptic Difference Equations , 1968 .

[20]  Philippe G. Ciarlet,et al.  The Finite Element Method for Elliptic Problems. , 1981 .

[21]  R. Bank,et al.  The hierarchical basis multigrid method , 1988 .

[22]  Wolfgang Hackbusch,et al.  Multi-grid methods and applications , 1985, Springer series in computational mathematics.

[23]  Ricardo G. Durán,et al.  On the asymptotic exactness of error estimators for linear triangular finite elements , 1991 .

[24]  Peter Deuflhard,et al.  Concepts of an adaptive hierarchical finite element code , 1989, IMPACT Comput. Sci. Eng..

[25]  Randolph E. Bank,et al.  A posteriori error estimates based on hierarchical bases , 1993 .

[26]  R. Bank,et al.  Some a posteriori error estimators for elliptic partial differential equations , 1985 .

[27]  Rüdiger Verfürth,et al.  A posteriori error estimation and adaptive mesh-refinement techniques , 1994 .

[28]  Alan Weiser,et al.  Local-mesh, local-order, adaptive finite element methods with a-posteriori error estimators for elliptic partial differential equations , 1981 .

[29]  Randolph E. Bank,et al.  Analysis Of A Two-Level Scheme For Solving Finite Element Equations , 1980 .

[30]  Maria Elizabeth G. Ong,et al.  Hierarchical Basis Preconditioners in Three Dimensions , 1997, SIAM J. Sci. Comput..

[31]  Harry Yserentant,et al.  A basic norm equivalence for the theory of multilevel methods , 1993 .

[32]  J. Maître,et al.  The contraction number of a class of two-level methods; an exact evaluation for some finite element subspaces and model problems , 1982 .

[33]  D. Braess.  The contraction number of a multigrid method for solving the Poisson equation , 1981 .

[34]  Peter Oswald,et al.  Multilevel Finite Element Approximation , 1994 .

[35]  H. Yserentant.  On the multi-level splitting of finite element spaces , 1986 .

[36]  I. Babuska,et al.  A‐posteriori error estimates for the finite element method , 1978 .

[37]  J. Whiteman.  The Mathematics of Finite Elements and Applications. , 1983 .

[38]  Harry Yserentant,et al.  On the multi-level splitting of finite element spaces , 1986 .

[39]  Ullrich Rüde.  Mathematical and Computational Techniques for Multilevel Adaptive Methods , 1987 .