Preconditioning of the p-version of the finite element method

The p-version finite element method for linear, second-order elliptic equations in an arbitrary, sufficiently smooth (incl. polygonal), bounded domain is studied in the framework of the Domain Decomposition (DD) method. Two types of square reference elements are used with coordinate functions given by the products of the integrated Legendre polynomials. Estimates for the condition numbers and some useful inequalities are given. We consider preconditioning of the problems arising on subdomains and of the Schur complement, as well as the derivation and analysis of the DD preconditioner for the entire system. This is done for a class of curvilinear finite elements. We obtain several DD preconditioners for which the generalized condition numbers vary from O((log p)3) to O(1). This paper is based on [19–21,27]. We have omitted most of the proofs in order to shorten it and have described instead what could be done as well as outlined some additional ideas. The full proofs omitted can in most cases be found in [19,20,27].

[1]  Yvon Maday,et al.  Polynomial interpolation results in Sobolev spaces , 1992 .

[2]  S. V. NEPOMNYASCHIKH.  Method of splitting into subspaces for solving elliptic boundary value problems in complex-form domains , 1991 .

[3]  Graham F. Carey,et al.  Basis function selection and preconditioning high degree finite element and spectral methods , 1989 .

[4]  Mark Ainsworth,et al.  A Preconditioner Based on Domain Decomposition for H-P Finite-Element Approximation on Quasi-Uniform Meshes , 1996 .

[5]  Claudio Canuto,et al.  Stabilization of spectral methods by finite element bubble functions , 1994 .

[6]  Mark M. Meerschaert,et al.  Mathematical Modeling , 2014, Encyclopedia of Social Network Analysis and Mining.

[7]  Anthony T. Patera,et al.  Analysis of Iterative Methods for the Steady and Unsteady Stokes Problem: Application to Spectral Element Discretizations , 1993, SIAM J. Sci. Comput..

[8]  I. Babuska,et al.  Efficient preconditioning for the p -version finite element method in two dimensions , 1991 .

[9]  M. Dryja,et al.  A finite element — Capacitance method for elliptic problems on regions partitioned into subregions , 1984 .

[10]  J. Pasciak,et al.  The Construction of Preconditioners for Elliptic Problems by Substructuring. , 2010 .

[11]  M. Deville,et al.  Preconditioned Chebyshev collocation methods and triangular finite elements , 1994 .

[12]  Michel O. Deville,et al.  Finite-Element Preconditioning for Pseudospectral Solutions of Elliptic Problems , 1990, SIAM J. Sci. Comput..

[13]  Ivo Babuška,et al.  The Problem of Selecting the Shape Functions for a p-Type Finite Element , 1989 .

[14]  Luca F. Pavarino,et al.  Schwarz methods with local refinement for the p-version finite element method , 1994 .

[15]  S. Jensen.  p -version of mixed finite element methods for Stokes-like problems , 1992 .

[16]  Jan Mandel,et al.  Iterative solvers by substructuring for the p -version finite element method , 1990 .

[17]  W. Han,et al.  On the sharpness of L 2 -error estimates of H 1 0 -projections onto subspaces of piecewise, high-order polynomials , 1995 .

[18]  Ernest E. Rothman,et al.  Preconditioning Legendre spectral collocation approximations to elliptic problems , 1995 .

[19]  J. Pasciak,et al.  Computer solution of large sparse positive definite systems , 1982 .