×

Preconditioning and a posteriori error estimates using \(h\)- and \(p\)-hierarchical finite elements with rectangular supports. (English) Zbl 1224.65077

The paper concentrates on two-dimensional linear elliptic problems with anisotropic diffusion coefficient and with homogeneous Dirichlet boundary conditions. The problem is discretized by the finite element method (FEM) with bilinear and higher-order rectangular elements. The main contribution is the computation of the uniform estimates of the constant in the strengthened Cauchy-Buniakowski-Schwarz inequality (CBS constant) for \(h\)- and \(p\)-hierarchies of rectangular finite elements. The CBS constant is used for showing optimality of the algebraic multilevel iterative (AMLI) preconditioner and for proving the efficiency of the hierarchical error estimates. A robust preconditioner for both \(h\)- and \(p\)-hierarchy is constructed as well as a new hierarchical a posteriori error estimate yielding lower bound on the energy norm of the error. The paper is concluded by illustrative numerical experiments.

MSC:

65F08 Preconditioners for iterative methods
65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
65N15 Error bounds for boundary value problems involving PDEs
35J25 Boundary value problems for second-order elliptic equations
65N55 Multigrid methods; domain decomposition for boundary value problems involving PDEs
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Axelsson, Two simple derivations of universal bounds for the C.B.S. inequality constant, Applications of Mathematics 49 pp 57– (2004) · Zbl 1099.65103
[2] Blaheta, Robust optimal preconditioners for non-conforming finite element systems, Numerical Linear Algebra with Applications 12 pp 495– (2005) · Zbl 1164.65518
[3] Axelsson, Preconditioning and two-level multigrid methods of arbitrary degree of approximations, Mathematics of Computation 40 pp 219– (1983) · Zbl 0511.65079
[4] Axelsson O, Blaheta R, Neytcheva M. Preconditioning of boundary value problems using elementwise Schur complement. Technical Report Series, Department of Information Technology, Uppsala University, 2006. · Zbl 1194.65047
[5] Georgiev I, Kraus J, Margenov S. Multilevel preconditioning of rotated bilinear non-conforming FEM problems. Technical Report, Johann Radon Institute for Computational and Applied Mathematics, January 2006. · Zbl 1142.65323
[6] Jung M, Maitre JF. Some remarks on the constant in the strengthened C.B.S. inequality: application to h- and p-hierarchical finite element discretizations of elasticity problems. Preprint SFB393/97-15, Technische Universität Chemnitz, 1997.
[7] Maitre, Multigrid Methods pp 535– (1982)
[8] Pultarová, Strengthened C.B.S. inequality constant for second order elliptic partial differential operator and for hierarchical bilinear finite element functions, Applications of Mathematics 50 pp 323– (2005)
[9] Axelsson, On the additive version of the algebraic multilevel iteration method for anisotropic elliptic problems, SIAM Journal on Scientific Computing 20 pp 1807– (1999) · Zbl 0940.65034
[10] Axelsson, Iterative Solution Methods (1996)
[11] Varga, Matrix Iterative Analysis (2000) · Zbl 1216.65042 · doi:10.1007/978-3-642-05156-2
[12] Bank, A posteriori error estimates based on hierarchical bases, SIAM Journal on Numerical Analysis 30 pp 921– (1993) · Zbl 0787.65078
[13] Šolín, Numerical Mathematics and Advanced Applications pp 683– (2006)
[14] Adjerid, Hierarchical finite element bases for triangular and tetrahedral elements, Computer Methods in Applied Mechanics and Engineering 190 pp 2925– (2001) · Zbl 0976.65102
[15] Lin, Superconvergence and extrapolation of non-conforming low order finite elements applied to the Poisson equation, IMA Journal of Numerical Analysis 25 pp 160– (2005) · Zbl 1068.65122
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.