The finite element method for a boundary value problem with strong singularity

Rukavishnikov, V. A.; Rukavishnikova, Elena I.

doi:10.1016/j.cam.2010.01.020

Cited by 42 publications

(19 citation statements)

References 17 publications

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“…Further we study the convergence of the approximate solution u h ν (x) to the exact solution u ν (x) in the norm of H 1 2,ν+β /2 (Ω) (see [30,35]). Further we study the convergence of the approximate solution u h ν (x) to the exact solution u ν (x) in the norm of H 1 2,ν+β /2 (Ω) (see [30,35]).…”

Section: Finite Element Methods In Its H H H-versionmentioning

confidence: 99%

“…Further we study the convergence of the approximate solution u h ν (x) to the exact solution u ν (x) in the norm of H 1 2,ν+β /2 (Ω) (see [30,35]). Moreover, by Lemma 2 from [35] we have (ρ ν+β /2 (x)u ν (x)) * Note that if the function u ν (x) ∈ H 2 2,ν+β /2 (Ω), then by Lemma 1 from [22], ρ ν+β /2 (x)u ν (x) belongs to W 2 2 (Ω).…”

Section: Finite Element Methods In Its H H H-versionmentioning

confidence: 99%

“…For problem (1.5)-(1.14) we formulate here the following theorems and corollaries (see, e.g., [35,38]). …”

Section: Boundary Value Problems With Coordinated Degeneration Of Inimentioning

confidence: 99%

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Methods of numerical analysis for boundary value problems with strong singularity

Rukavishnikov

2009

Russian Journal of Numerical Analysis and Mathematical Modelling

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Numerical methods and studies of differential properties of an R ν -weak solution are presented for boundary value problems with a divergent Dirichlet integral of the solution.In this paper we discuss numerical methods and studies of differential properties of boundary value problems for elliptic equations possessing strong singularity. A boundary value problem is said to possess strong singularity if its solution u(x) does not belong to the Sobolev space W 1 2 H 1 or, in other words, the Dirichlet integral of the solution u(x) diverges.There exists a complete theory of classic solutions to boundary value problems with smooth initial data (coefficients of equations, right-hand sides of equations, boundary conditions) and a sufficiently smooth domain boundary (see, e.g., [1,11,13,17]). A grid method for boundary value problems with singularity of the solution caused by the angles of the domain boundary was first considered by Volkov [48].Based on the notion of a weak solution, extensive studies of boundary value problems with discontinuous initial data and nonsmooth domain boundaries were performed for Sobolev and various weighted spaces (see, e.g., [10,12,15,16,18]). Based on the Galerkin approach, theories of difference schemes, finite volumes, and finite element methods were elaborated for determination of approximate weak solutions (see, e.g., [8,45,46] and others).Boundary value problems with strong singularity caused by the singularity in the initial data or by the internal properties of the solution are found in the physics of plasma and gas discharge, electrodynamics, nuclear physics, nonlinear optics, and other branches of physics. In particular cases, numerical methods for problems of electrodynamics and quantum mechanics with strong singularity, were constructed based on separation of singular and regular components, mesh refinement near singular points, multiplicative extraction of singularities, etc. (see, e.g., [2,3,9,47]).In [21] it was proposed to define the solution to a boundary value problem with strong singularity as an R ν -weak solution in a weighted Sobolev space. This new notion of the solution allows one to discriminate two classes of boundary value *

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Section: Finite Element Methods In Its H H H-versionmentioning

confidence: 99%

Section: Finite Element Methods In Its H H H-versionmentioning

confidence: 99%

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Methods of numerical analysis for boundary value problems with strong singularity

Rukavishnikov

2009

Russian Journal of Numerical Analysis and Mathematical Modelling

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“…For boundary value problems with strongly singular solutions we developed the theory of numerical methods based on the conception of R ν -generalized solution (see, for example, [14]- [18]). This conception allows us, depending on the singularity of input data (coefficients and right hand sides of equations and boundary conditions) and geometry of the boundary, to define a weighted space or a set containing the solution.…”

Section: Introductionmentioning

confidence: 99%

The weighted edge finite element method for time-harmonic maxwell equations with strong singularity

Rukavishnikov

Mosolapov

2014

2014 International Conference on Mathematical Methods in Electromagnetic Theory

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“…The existence and uniqueness of solutions as well as its coercivity and differential properties in the weighted Sobolev spaces and sets were proved [5][6][7][8][9][10], the weighted finite-element method was built, and its convergence rate was investigated [11][12][13][14][15].…”

mentioning

confidence: 99%

Weighted Finite-Element Method for Elasticity Problems with Singularity

Rukavishnikov¹,

IvanovnaRukavishnikova²

2018

Finite Element Method - Simulation, Numerical Analysis and Solution Techniques

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In this chapter, the two-dimensional elasticity problem with a singularity caused by the presence of a re-entrant corner on the domain boundary is considered. For this problem, the notion of the R v -generalized solution is introduced. On the basis of the R v -generalized solution, a scheme of the weighted finite-element method (FEM) is constructed. The proposed method provides a first-order convergence of the approximate solution to the exact one with respect to the mesh step in the W 1 2, ν Ω ðÞ -norm. The convergence rate does not depend on the size of the angle and kind of the boundary conditions imposed on its sides. Comparative analysis of the proposed method with a classical finite-element method and with an FEM with geometric mesh refinement to the singular point is carried out.

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The finite element method for a boundary value problem with strong singularity

Cited by 42 publications

References 17 publications

Methods of numerical analysis for boundary value problems with strong singularity

Methods of numerical analysis for boundary value problems with strong singularity

The weighted edge finite element method for time-harmonic maxwell equations with strong singularity

Weighted Finite-Element Method for Elasticity Problems with Singularity

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