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Symmetry
Special Issues
Mesh Methods—Numerical Analysis and Experiments II
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Mesh Methods—Numerical Analysis and Experiments II

A special issue of Symmetry (ISSN 2073-8994). This special issue belongs to the section "Mathematics".

Deadline for manuscript submissions: closed (31 December 2023) | Viewed by 3944

Special Issue Editor

Prof. Dr. Viktor A. Rukavishnikov

E-Mail Website
Guest Editor
Computing Center of Far-Eastern Branch, Russian Academy of Sciences, Kim-Yu-Chen Str. 65, Khabarovsk 680000, Russia
Interests: boundary value problems with singularity; numerical methods in electrodynamics; hydrodynamics and theory of elasticity
Special Issues, Collections and Topics in MDPI journals

Special Issue Information

Dear Colleagues,

Mathematical models of different natural processes are described by the differential equations, systems of PDEs and integral equations. In most cases, it turns out that the exact solution of such problems cannot be determined, so we have to use mesh methods to calculate approximate solutions using high-performance computational complexes. First of all, these methods include the finite element method, the finite difference method, the finite volume method and the combined methods.

In this Special Issue, it is proposed to publish qualitative works on theoretical studies of grid methods on the approximation, stability, and convergence, as well as the results of numerical experiments confirming the effectiveness of the developed methods. New methods for boundary value problems with singularity, with the complex geometry of the domain boundary and for non-linear equations are under particular interest. Articles concerning the analysis of the numerical methods developed for the computation of the mathematical models in different areas of applied science and engineering applications will be welcome.

As a rule, symmetry ideas are present in the computational schemes and make the process harmonic and effective.

Prof. Dr. Viktor A. Rukavishnikov
Guest Editor

Manuscript Submission Information

Manuscripts should be submitted online at www.mdpi.com by registering and logging in to this website. Once you are registered, click here to go to the submission form. Manuscripts can be submitted until the deadline. All submissions that pass pre-check are peer-reviewed. Accepted papers will be published continuously in the journal (as soon as accepted) and will be listed together on the special issue website. Research articles, review articles as well as short communications are invited. For planned papers, a title and short abstract (about 100 words) can be sent to the Editorial Office for announcement on this website.

Submitted manuscripts should not have been published previously, nor be under consideration for publication elsewhere (except conference proceedings papers). All manuscripts are thoroughly refereed through a single-blind peer-review process. A guide for authors and other relevant information for submission of manuscripts is available on the Instructions for Authors page. Symmetry is an international peer-reviewed open access monthly journal published by MDPI.

Please visit the Instructions for Authors page before submitting a manuscript. The Article Processing Charge (APC) for publication in this open access journal is 2400 CHF (Swiss Francs). Submitted papers should be well formatted and use good English. Authors may use MDPI's English editing service prior to publication or during author revisions.

Keywords

  • finite element method
  • difference method
  • finite volume method
  • numerical experiments
  • numerical analysis
  • corner singularity
  • symmetry

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Published Papers (2 papers)

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Research

15 pages, 1224 KiB  
Article
The Numerical Solution of Large-Scale Generalized Eigenvalue Problems Arising from Finite-Element Modeling of Electroelastic Materials
by Tatiana Martynova, Galina Muratova, Pavel Oganesyan and Olga Shtein
Symmetry 2023, 15(1), 171; https://doi.org/10.3390/sym15010171 - 6 Jan 2023
Cited by 2 | Viewed by 1952
Abstract
The generalized eigenvalue problem for a symmetric definite matrix pencil obtained from finite-element modeling of electroelastic materials is solved numerically by the Lanczos algorithm. The mass matrix is singular in the considered problem, and therefore the process proceeds with the semi-inner product defined [...] Read more.
The generalized eigenvalue problem for a symmetric definite matrix pencil obtained from finite-element modeling of electroelastic materials is solved numerically by the Lanczos algorithm. The mass matrix is singular in the considered problem, and therefore the process proceeds with the semi-inner product defined by this matrix. The shift-and-invert Lanczos algorithm is used to find multiple eigenvalues closest to some shift and the corresponding eigenvectors. The results of the numerical experiments are presented. Full article
(This article belongs to the Special Issue Mesh Methods—Numerical Analysis and Experiments II)
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12 pages, 21891 KiB  
Article
On the Error Estimation of the FEM for the Nikol’skij-Lizorkin Problem with Degeneracy in the Lebesgue Space
by Viktor Rukavishnikov and Elena Rukavishnikova
Symmetry 2022, 14(6), 1276; https://doi.org/10.3390/sym14061276 - 20 Jun 2022
Viewed by 1383
Abstract
In this manuscript, the Nikol’skii-Lizorkin problem with degeneracy on the entire boundary of the domain is considered. The degeneracy is caused by the behavior of the coefficients and right-hand sides of the problem on the boundary. The triangulation of the domain with a [...] Read more.
In this manuscript, the Nikol’skii-Lizorkin problem with degeneracy on the entire boundary of the domain is considered. The degeneracy is caused by the behavior of the coefficients and right-hand sides of the problem on the boundary. The triangulation of the domain with a special compression to the boundary of the domain is carried out and the finite element method is constructed. An estimate for the rate of convergence of an approximate solution to an exact solution with the second order in the mesh step in the L2(Ω) space norm is proved. Numerical experiments have confirmed the established estimate of the convergence rate. Full article
(This article belongs to the Special Issue Mesh Methods—Numerical Analysis and Experiments II)
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