On finite element methods for elliptic equations on domains with corners

Blum, H.; Dobrowolski, Manfred

doi:10.1007/bf02237995

Cited by 109 publications

(75 citation statements)

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“…A number of methods have been proposed for computing the constants of asymptotic expansion (called "stress intensity factors" in elasticity). Most of the works considered only linear problems [2,7,10,11,19,21,26,33,35]. Blum [6] discussed application of dual singular function method to semilinear biharmonic equations, such as the Navier-Stokes equations or the von Kármán equations, however, presenting the numerical results only for linear problems.…”

Section: Introductionmentioning

confidence: 99%

“…It has been found out that local mesh refinement near corners and use of analytical formulas of asymptotic solution near corners produce better results for problems with corner singularities. Some of the popular techniques to overcome slow convergence are as follows: singular function method [19,35], singular complement method [2], dual singular function method [6,7,10,11], introduction of analytical constraints to finite element formulation [33], truncation of corners and introduction of Dirichlet-to-Neumann boundary conditions for domains with truncated corners [21], and other methods based on the similar ideas [26,34]. Also, various methods based solely on mesh refinement (without using asymptotic expansion of the solution) were developed (see, for example, [1,3,14,30,31]).…”

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confidence: 99%

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An Asymptotic Fitting Finite Element Method with Exponential Mesh Refinement for Accurate Computation of Corner Eddies in Viscous Flows

Shapeev¹,

Lin²

2009

SIAM J. Sci. Comput.

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General rights Copyright and moral rights for the publications made accessible in Discovery Research Portal are retained by the authors and/or other copyright owners and it is a condition of accessing publications that users recognise and abide by the legal requirements associated with these rights.• Users may download and print one copy of any publication from Discovery Research Portal for the purpose of private study or research.• You may not further distribute the material or use it for any profit-making activity or commercial gain.• You may freely distribute the URL identifying the publication in the public portal. Take down policyIf you believe that this document breaches copyright please contact us providing details, and we will remove access to the work immediately and investigate your claim. Abstract.It is well known that any viscous fluid flow near a corner consists of infinite series of eddies with decreasing size and intensity, unless the angle is larger than a certain critical angle [H. K. Moffat, J. Fluid Mech., 18 (1964), pp. 1-18]. The objective of the current work is to simulate such infinite series of eddies occurring in steady flows in domains with corners. The problem is approached by high-order finite element method with exponential mesh refinement near the corners, coupled with analytical asymptotics of the flow near the corners. Such approach allows one to compute position and intensity of the eddies near the corners in addition to the other main features of the flow. The method was tested on the problem of the lid-driven cavity flow as well as on the problem of the backward-facing step flow. The results of computations of the lid-driven cavity problem show that the proposed method computes the central eddy with accuracy comparable to the best of existing methods and is more accurate for computing the corner eddies than the existing methods. The results also indicate that the relative error of finding the eddies' intensity and position decreases uniformly for all the eddies as the mesh is refined (i.e., the relative error in computation of different eddies does not depend on their size). 1. Introduction. The two-dimensional flow of a viscous fluid near the corner between two steady rigid planes was first examined by Moffatt [28]. He established that when the angle between planes is less than a certain critical angle, any flow near the corner consists of infinite series of eddies with decreasing size and intensity as the corner point is approached.One of the most famous examples of flow in domain with corners is a flow in the lid-driven cavity. The lid-driven cavity problem has become a benchmark problem for researchers to test the performance of numerical methods designed for computation of viscous fluid flow. Particularly, among other criteria, the researchers examine the accuracy of their methods based on how accurately they can compute the corner eddies. However, in the previous works only a few eddies were computed (maximum four corner eddies [4,18] for certain Reynolds numbers). In ad...

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Section: Introductionmentioning

confidence: 99%

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confidence: 99%

An Asymptotic Fitting Finite Element Method with Exponential Mesh Refinement for Accurate Computation of Corner Eddies in Viscous Flows

Shapeev¹,

Lin²

2009

SIAM J. Sci. Comput.

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“…To achieve optimal convergence of lower order finite elements on quasi-uniform grids, a class of finite element methods studied by various researchers is to make use of the singular function representation of the solution. This includes the singular function method (see, e.g., [10,8]), the dual singular function method (see [9,2,3]), and the multigrid version of the dual singular function method (see [4]). …”

Section: Introductionmentioning

confidence: 99%

A Finite Element Method Using Singular Functions for the Poisson Equation: Corner Singularities

Cai

Kim

2001

SIAM J. Numer. Anal.

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The solution of the interface problem is only in H 1+α (Ω) with α > 0 possibly close to zero and, hence, it is difficult to be approximated accurately. This paper studies an accurate numerical method on quasi-uniform grids for two-dimensional interface problems. The method makes use of a singular function representation of the solution, dual singular functions, and an extraction formula for stress intensity factors. Using continuous piecewise linear elements on quasi-uniform grids, our finite element approximation is shown to be optimal, O(h), accurate in the H 1 norm. This is confirmed by numerical experiments for interface problems with α < 0.1. An O(h 1+α ) error bound in the L 2 norm is also established by the standard duality argument. For small α, this improvement over the H 1 error bound is negligible. However, numerical tests presented in this paper indicate that the L 2 norm accuracy is much better than the theoretical error bound.

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“…Thus, the EFIFs cannot be obtained in a straightforward manner over this twodimensional plane, and special methods for their computation have to be used. It is important to note that straightforward implementation of 2-D e cient extraction methods for the EFIFs as the contour integral method (also known as the dual singular function method [10]), or the cuto function method [8], are not possible either, and will lead to false results unless a proper treatment is incorporated. We present a special method based on L 2 projection and Richardson extrapolation for point-wise extraction of EFIFs from p-ÿnite element solutions.…”

Section: Discussionmentioning

confidence: 99%

Extracting edge flux intensity functions for the Laplacian

Yosibash

Actis

Szabó

2001

Numerical Meth Engineering

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SUMMARYThe solution to the Laplace operator in three-dimensional domains in the vicinity of straight edges is presented as an asymptotic expansion involving eigenpairs with their coe cients called edge ux intensity functions (EFIFs). The eigenpairs are identical to their two-dimensional counterparts over a plane perpendicular to the edge. Extraction of EFIFs however, cannot be obtained in a straightforward manner over this two-dimensional plane. A method based on L 2 projection and Richardson extrapolation is presented for point-wise extraction of EFIFs from p-ÿnite element solutions and illustrated by examples. A similar but more e cient method, based on 'energy projection', for extracting EFIFs is proposed.

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On finite element methods for elliptic equations on domains with corners

Cited by 109 publications

References 2 publications

An Asymptotic Fitting Finite Element Method with Exponential Mesh Refinement for Accurate Computation of Corner Eddies in Viscous Flows

An Asymptotic Fitting Finite Element Method with Exponential Mesh Refinement for Accurate Computation of Corner Eddies in Viscous Flows

A Finite Element Method Using Singular Functions for the Poisson Equation: Corner Singularities

Extracting edge flux intensity functions for the Laplacian

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