Higher order meshless schemes applied to the finite element method in elliptic problems
Authors:
- Sławomir Milewski,
- Roman Putanowicz
Abstract
This paper presents selected approximation techniques, typical for the meshless finite difference method (MFDM), although applied to the finite element method (FEM). Finite elements with standard or hierarchical shape functions are coupled with higher order meshless schemes, based upon the correction terms of a simple difference operator. Those terms consist of higher order derivatives, which are evaluated by means of the appropriate formulas composition as well as a numerical solution, which corresponds to the primary interpolation order, assigned to element shape functions. Correction terms modify the right-hand sides of algebraic FE equations only, yielding an iterative procedure. Therefore, neither re-generation of the stiffness matrix nor introduction of any additional nodes and/or degrees of freedom is required. Such improved FE-MFD solution approach allows for the optimal application of advantages of both methods, for instance, a high accuracy of the nodal FE solution and a derivatives’ super-convergence phenomenon at arbitrary domain points, typical for the meshless FDM. Existing and proposed higher order techniques, applied in the FEM, are compared with each other in terms of the solution accuracy, algorithm efficiency and computational complexity. In order to examine the considered algorithms, numerical results of several two-dimensional benchmark elliptic problems are presented. Both the accuracy of a solution and the solution’s derivatives as well as their convergence rates, evaluated on irregular and structured meshes as well as arbitrarily irregular adaptive clouds of nodes, are taken into account.
- Record ID
- CUTbb63b0cbc3454906b73b3253ccd1aa15
- Publication categories
- ;
- Author
- Journal series
- Computers & Mathematics With Applications, ISSN 0898-1221, e-ISSN 1873-7668
- Issue year
- 2019
- Vol
- 77
- No
- 3
- Pages
- 779-802
- Other elements of collation
- rys.; tab.; wykr.; Bibliografia (na s.) - 801-802; Bibliografia (liczba pozycji) - 76; Oznaczenie streszczenia - Abstr.; Data udostępnienia on-line - 2018-10-26; Numeracja w czasopiśmie - Vol. 77, Iss. 3
- Keywords in English
- finite element method, meshless methods, meshless finite difference method, higher order approximation, elliptic problems
- DOI
- DOI:10.1016/j.camwa.2018.10.016 Opening in a new tab
- URL
- https://www.sciencedirect.com/science/article/pii/S0898122118306138 Opening in a new tab
- Related project
- Komputerowa analiza inżynierskich nieustalonych zagadnień termosprężystych i termoplastycznych za pomocą sprzężonej MES / bezsiatkowej MRS. . Project leader at PK: , ,
Projects financed by NSC [Projekty finansowane przez NCN] - Language
- eng (en) English
- Score (nominal)
- 140
- Additional fields
- Indeksowana w: Web of Science, Scopus
- Uniform Resource Identifier
- https://cris.pk.edu.pl/info/article/CUTbb63b0cbc3454906b73b3253ccd1aa15/
- URN
urn:pkr-prod:CUTbb63b0cbc3454906b73b3253ccd1aa15
* presented citation count is obtained through Internet information analysis, and it is close to the number calculated by the Publish or PerishOpening in a new tab system.