The strengthened C.B.S. inequality constant for second order elliptic partial differential operator and for hierarchical bilinear finite element functions

Pultarová, Ivana

doi:10.1007/s10492-005-0020-4

Cited by 7 publications

(7 citation statements)

References 6 publications

(10 reference statements)

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“…In the former case, γ 2 < 3/4 for all triangles, even for degenerate ones (when one or two angles equal zero). The constant for an anisotropic operator for bilinear elements on a rectangular mesh is also γ 2 < 3/4, see [20] for a derivation.…”

Section: This Implies Vmentioning

confidence: 99%

Preconditioning of Boundary Value Problems Using Elementwise Schur Complements

Axelsson¹,

Blaheta²,

Neytcheva³

2009

SIAM J. Matrix Anal. & Appl.

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Based on a particular node ordering and corresponding block decomposition of the matrix we analyse an ecient, algebraic multilevel preconditioner for the iterative solution of nite element discretizations of elliptic boundary value problems. Thereby an analysis of a new version of block-factorization preconditioning methods is presented. The approximate factorization requires an approximation of the arising Schur complement matrix. In this paper we consider such approximations derived by the assembly of the local macro-element Schur complements. The method can be applied also for non-selfadjoint problems but for the derivation of condition number bounds we assume that the corresponding dierential operator is selfadjoint and positive denite.

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Section: This Implies Vmentioning

confidence: 99%

Preconditioning of Boundary Value Problems Using Elementwise Schur Complements

Axelsson¹,

Blaheta²,

Neytcheva³

2009

SIAM J. Matrix Anal. & Appl.

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“…However, the approximation used for the Schur complement in this paper leads to a dependence on γ, see the estimate in Section 3. As has been shown in [7,1,23], for instance, for problems with constant, or smoothly varying coefficients in each element, γ is bounded uniformly away from its critical value unity, even for arbitrary degree of unisotropy and jumps of coefficients between elements. For jumps in the interior of an element, however, it has been shown in [17] that the parameter γ can take values arbitrary close to unity when the ratio of coefficients increase.…”

Section: Introductionmentioning

confidence: 81%

Macro-elementwise preconditioning methods

Axelsson

2012

Mathematics and Computers in Simulation

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“…nonuniform grid and bilinear FEs, we can get 2 ≤ 3 4 , cf. [4,35]. It is important that a sharp uniform upper bound to the CBS constant can be obtained by restricting (38) to a single coarse element independently of the discretization density, see e.g.…”

Section: Discussionmentioning

confidence: 99%

“…Lemma 7.1 Let the matrix A be the stiffness matrix obtained for the operator (35) and for the bilinear FE functions. Let us consider a two-level splitting with properties (36) and (37).…”

Section: Discussionmentioning

confidence: 99%

“…Having obtained u and v , as the coefficient vectors of u and v, respectively, we get where Note that all three matrices A ML , A ML,C , and A ML,F are positive definite. In fact A ML,C and A ML,F can be obtained by the restriction of (35) to V N C and W N C , respectively. Since V N C and W N C are finite-dimensional and the only common function of V N C and W N C is the zero function, there exists ∈ ⟨0, 1) such that for all v ∈ ℝ N .…”

Section: Example 61mentioning

confidence: 99%

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Multilevel a posteriori error estimator for greedy reduced basis algorithms

2020

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The goal of reduced basis algorithms is to provide a relatively small set of functions which can serve as a basis for sufficiently accurate and fast numerical solution of a parametrized problem for any choice of parameters. Such methods are often employed in parameter identification problems detecting, for instance, material qualities in diffusion problems, elasticity, or in Maxwell equations, or in time dependent problems where time plays the role of the parameter. We deal with greedy reduced basis algorithms. An important part of these algorithms is to estimate the difference between the exact solution of a discretized problem and its projection onto the space spanned by a reduced basis. We introduce a new kind of the estimate, which is based on a multilevel splitting of a discretized solution space, and we compare it with a standard estimate based on bounds to coercivity and continuity constants. Two sided guaranteed bounds to the error can be obtained for both methods. Numerical complexity as well as memory consumption of both methods are comparable, while the multilevel method provides a more accurate spatial distribution of the error.

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The strengthened C.B.S. inequality constant for second order elliptic partial differential operator and for hierarchical bilinear finite element functions

Cited by 7 publications

References 6 publications

Preconditioning of Boundary Value Problems Using Elementwise Schur Complements

Preconditioning of Boundary Value Problems Using Elementwise Schur Complements

Macro-elementwise preconditioning methods

Multilevel a posteriori error estimator for greedy reduced basis algorithms

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