A quadrature finite element Galerkin scheme for a biharmonic problem on a rectangular polygon

Aitbayev, Rakhim

doi:10.1002/num.20278

Cited by 4 publications

(10 citation statements)

References 10 publications

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“…The methods and results in this article are closely related to those in [5][6][7][8][9][10]. The quadrature Galerkin problem of the present article is studied in [6] for existence, uniqueness, and convergence of a solution.…”

Section: Introductionmentioning

confidence: 94%

“…where u is the solution of problem (1.2), and H −2+2s ( ) is the dual of a fractional order Sobolev space (Theorem 3.1 in [6]). …”

Section: Quadrature Schemementioning

confidence: 99%

“…The quadrature Galerkin problem of the present article is studied in [6] for existence, uniqueness, and convergence of a solution. The optimal order error estimates in Sobolev norms are obtained under a sufficient regularity assumption using the fact that the quadrature Galerkin problem is equivalent to an orthogonal spline collocation scheme, which is related to that in [9].…”

Section: Introductionmentioning

confidence: 99%

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Multilevel preconditioners for a quadrature Galerkin solution of a biharmonic problem

Aitbayev

2006

Numer. Methods Partial Differential Eq.

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Efficient multilevel preconditioners are developed and analyzed for the quadrature finite element Galerkin approximation of the biharmonic Dirichlet problem. The quadrature scheme is formulated using the BognerFox-Schmit rectangular element and the product two-point Gaussian quadrature. The proposed additive and multiplicative preconditioners are uniformly spectrally equivalent to the operator of the quadrature scheme. The preconditioners are implemented by optimal algorithms, and they are used to accelerate convergence of the preconditioned conjugate gradient method. Numerical results are presented demonstrating efficiency of the preconditioners.

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Section: Introductionmentioning

confidence: 94%

“…where u is the solution of problem (1.2), and H −2+2s ( ) is the dual of a fractional order Sobolev space (Theorem 3.1 in [6]). …”

Section: Quadrature Schemementioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Multilevel preconditioners for a quadrature Galerkin solution of a biharmonic problem

Aitbayev

2006

Numer. Methods Partial Differential Eq.

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“…The finite element Galerkin method for biharmonic problems based on a weak formulation requires an H 2 trial space, see the recent work [1,2] and references therein. A byproduct of the analysis in this work in Section 3 yields optimal order convergence of a quadrature finite element Galerkin solution for a nonlinear biharmonic problem, extending the analysis in [2]. Further, the scheme in this paper has the advantage of allowing a wider class of nonlinear processes in the advectiondiffusion-reaction model (1.1), (1.2) and (1.3).…”

Section: Introductionmentioning

confidence: 99%

A fully discrete H 1-Galerkin method with quadrature for nonlinear advection–diffusion–reaction equations

Ganesh

Mustapha

2007

Numer Algor

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We propose and analyze a fully discrete H 1 -Galerkin method with quadrature for nonlinear parabolic advection-diffusion-reaction equations that requires only linear algebraic solvers. Our scheme applied to the special case heat equation is a fully discrete quadrature version of the least-squares method. We prove second order convergence in time and optimal H 1 convergence in space for the computer implementable method. The results of numerical computations demonstrate optimal order convergence of scheme in H k for k = 0, 1, 2.

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“…The advantages of using the former approach include better asymptotic accuracy for the same level of grid resolution (see [1,Thm. 5.4], [10, Thms.…”

Section: Introductionmentioning

confidence: 99%

Efficient Block Preconditioning for a $C^1$ Finite Element Discretization of the Dirichlet Biharmonic Problem

Pestana¹,

Muddle²,

Heil³

et al. 2016

SIAM J. Sci. Comput.

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Abstract. We present an efficient block preconditioner for the two-dimensional biharmonic Dirichlet problem discretized by C 1 bicubic Hermite finite elements. In this formulation each node in the mesh has four different degrees of freedom (DOFs). Grouping DOFs of the same type together leads to a natural blocking of the Galerkin coefficient matrix. Based on this block structure, we develop two preconditioners: a 2 × 2 block diagonal (BD) preconditioner and a block bordered diagonal (BBD) preconditioner. We prove mesh-independent bounds for the spectra of the BDpreconditioned Galerkin matrix under certain conditions. The eigenvalue analysis is based on the fact that the proposed preconditioner, like the coefficient matrix itself, is symmetric positive definite (SPD) and assembled from element matrices. We demonstrate the effectiveness of an inexact version of the BBD preconditioner, which exhibits near-optimal scaling in terms of computational cost with respect to the discrete problem size. Finally, we study robustness of this preconditioner with respect to element stretching, domain distortion, and nonconvex domains.

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A quadrature finite element Galerkin scheme for a biharmonic problem on a rectangular polygon

Cited by 4 publications

References 10 publications

Multilevel preconditioners for a quadrature Galerkin solution of a biharmonic problem

Multilevel preconditioners for a quadrature Galerkin solution of a biharmonic problem

A fully discrete H 1-Galerkin method with quadrature for nonlinear advection–diffusion–reaction equations

Efficient Block Preconditioning for a $C^1$ Finite Element Discretization of the Dirichlet Biharmonic Problem

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