Asymptotical expansion in non‐Lipschitzian domains – a numerical approach

Dimitrov, A.; Schnack, E.

doi:10.1002/nla.292

Cited by 8 publications

(14 citation statements)

References 21 publications

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“…The number λ is called eigenvalue of problem (14), and the vector function u is called eigenelement corresponding to λ.…”

Section: Function Spaces and Weak Formulation Of The Problemmentioning

confidence: 99%

“…They are also generalized eigenelements of the weakly formulated eigenvalue problem (14) to some eigenvalue λ 0 = α 0 + 1 2 and satisfy the following variational equations,…”

Section: Is Called Jordan Chain Of the Length K Of A(·) At α 0 If Thementioning

confidence: 99%

“…Assume that λ 0 is an eigenvalue of problem (14) and (λ h , u h ) are eigenpairs of problem (25), such that λ h → λ 0 as h → 0. Then for each element u h there exists an eigenelement u 0 = u 0 (h) ∈ U (B, λ 0 ) of problem (14) corresponding to λ 0 such that for sufficiently small h the following error estimate holds:…”

Section: Theorem 47mentioning

confidence: 99%

“…Then for each element u h there exists an eigenelement u 0 = u 0 (h) ∈ U (B, λ 0 ) of problem (14) corresponding to λ 0 such that for sufficiently small h the following error estimate holds:…”

Section: Theorem 47mentioning

confidence: 99%

“…• is also calculated in [14] by using an adaptive finite element method and by exploiting the symmetry in the eigensolutions. There, the approximate convergence order 2.12 was obtained for linear finite elements and 4.60 for quadratic elements.…”

Section: Approximation Error Estimatesmentioning

confidence: 99%

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Computation of 3D vertex singularities for linear elasticity: Error estimates for a finite element method on graded meshes

2002

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Abstract. This paper is concerned with the computation of 3D vertex singularities of anisotropic elastic fields with Dirichlet boundary conditions, focusing on the derivation of error estimates for a finite element method on graded meshes. The singularities are described by eigenpairs of a corresponding operator pencil on spherical polygonal domains. The main idea is to introduce a modified quadratic variational boundary eigenvalue problem which consists of two self-adjoint, positive definite sesquilinear forms and a skew-Hermitean form. This eigenvalue problem is discretized by a finite element method on graded meshes. Based on regularity results for the eigensolutions estimates for the finite element error are derived both for the eigenvalues and the eigensolutions. Finally, some numerical results are presented.Mathematics Subject Classification. 65N25, 65N30, 74G70.

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“…The number λ is called eigenvalue of problem (14), and the vector function u is called eigenelement corresponding to λ.…”

Section: Function Spaces and Weak Formulation Of The Problemmentioning

confidence: 99%

“…They are also generalized eigenelements of the weakly formulated eigenvalue problem (14) to some eigenvalue λ 0 = α 0 + 1 2 and satisfy the following variational equations,…”

Section: Is Called Jordan Chain Of the Length K Of A(·) At α 0 If Thementioning

confidence: 99%

Section: Theorem 47mentioning

confidence: 99%

Section: Theorem 47mentioning

confidence: 99%

Section: Approximation Error Estimatesmentioning

confidence: 99%

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Computation of 3D vertex singularities for linear elasticity: Error estimates for a finite element method on graded meshes

2002

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References

2017

Handbook of Structural Life Assessment

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On singularities in the solution of three‐dimensional Stokes flow and incompressible elasticity problems with corners

Dimitrov

2004

Numerical Meth Engineering

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SUMMARYIn this paper, a numerical procedure is presented for the computation of corner singularities in the solution of three-dimensional Stokes flow and incompressible elasticity problems near corners of various shape. For obtaining the order and mode of singularity, a neighbourhood of the singular point is considered with only local boundary conditions. The weak formulation of this problem is approximated using a mixed u, p Galerkin-Petrov finite element method. Additionally, a separation of variables is used to reduce the dimension of the original problem. As a result, the quadratic eigenvalue problem (P + Q + 2 R)d = 0 is obtained, where the saddle-point-type matrices P, Q, R are defined explicitly. For a numerical solution of the algebraic eigenvalue problem an iterative technique based on the Arnoldi method in combination with an Uzawa-like scheme is used. This technique needs only one direct matrix factorization as well as few matrix-vector products for finding all eigenvalues in the interval R( ) ∈ (−0.5, 1.0), as well as the corresponding eigenvectors. Some benchmark tests show that this technique is robust and very accurate. Problems from practical importance are also analysed, for instance the surface-breaking crack in an incompressible elastic material and the three-dimensional viscous flow of a Newtonian fluid past a trihedral corner.

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Asymptotical expansion in non‐Lipschitzian domains – a numerical approach

Cited by 8 publications

References 21 publications

Computation of 3D vertex singularities for linear elasticity: Error estimates for a finite element method on graded meshes

Computation of 3D vertex singularities for linear elasticity: Error estimates for a finite element method on graded meshes

References

On singularities in the solution of three‐dimensional Stokes flow and incompressible elasticity problems with corners

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