Article information

2016 , Volume 21, ¹ 6, p.89-103

Rukavishnikov V.A., Nikolaev S.G.

Numerical analysis of the weighted finite element method for the elasticity problem with singularity

In this paper we consider the first boundary value problem for the Lam´e system in the two-dimensional nonconvex domain with boundary containing one reentrant corner. For this problem we define the solution as 𝑅𝜈 -generalized one in the weighted set. On the basis of this solution we constructed a scheme of the weighted finite element method with accuracy independent of the corner size and with 𝑂(ℎ) convergence rate for the approximate solution to the exact one.

We carried out a comparative numerical analysis of the weighted finite element method, the classical FEM, and the FEM with meshes geometrically refined to the point of singularity. For this purpose we considered model problems for L-shaped domain. Singularity order of solutions of these problems was set as consistent with the magnitude 3𝜋/2 of the reentrant corner.

Performed numerical analysis has demonstrated that

- the approximate 𝑅𝜈 -generalized solution of the problem converges to the exact one at 𝑂(ℎ) rate in the weighted Sobolev norm, while the generalized solution (for the classical FEM) has 𝑂(ℎ 0.61 ) rate of convergence in the Sobolev norm;

- in the overwhelming majority of grid nodes, the absolute error of the approximate 𝑅𝜈 -generalized solution is one or two orders of magnitude lower than for the generalized solution;

- the FEM with graded meshes fails on high-dimensional grids, but the weighted FEM stably allows finding approximate solution with the high accuracy under the same computational conditions.

[full text]
Keywords: boundary value problem with singularity, weighted finite element method, R-generalized solution

Author(s):
Rukavishnikov Viktor Anatolyevich
Dr. , Professor
Position: Head of Laboratory
Office: Computig Center FEB RAS
Address: 680000, Russia, Khabarovsk, Kim U Chen st., 65
Phone Office: (4212)704342
E-mail: vark0102@mail.ru
SPIN-code: 5421-6907

Nikolaev Sergey Georgievich
Position: Research Scientist
Office: Computig Center FEB RAS
Address: 680000, Russia, Khabarovsk, Kim U Chen st., 65
Phone Office: (4212)704342
E-mail: snik-post@ya.ru
SPIN-code: 6250-1370

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Bibliography link:
Rukavishnikov V.A., Nikolaev S.G. Numerical analysis of the weighted finite element method for the elasticity problem with singularity // Computational technologies. 2016. V. 21. ¹ 6. P. 89-103
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