Very high order discontinuous Galerkin method in elliptic problems
Authors:
- Jan Jaśkowiec
Abstract
The paper deals with high-order discontinuous Galerkin (DG) method with the approximation order that exceeds 20 and reaches 100 and even 1000 with respect to one-dimensional case. To achieve such a high order solution, the DG method with finite difference method has to be applied. The basis functions of this method are high-order orthogonal Legendre or Chebyshev polynomials. These polynomials are defined in one-dimensional space (1D), but they can be easily adapted to two-dimensional space (2D) by cross products. There are no nodes in the elements and the degrees of freedom are coefficients of linear combination of basis functions. In this sort of analysis the reference elements are needed, so the transformations of the reference element into the real one are needed as well as the transformations connected with the mesh skeleton. Due to orthogonality of the basis functions, the obtained matrices are sparse even for finite elements with more than thousands degrees of freedom. In consequence, the truncation errors are limited and very high-order analysis can be performed. The paper is illustrated with a set of benchmark examples of 1D and 2D for the elliptic problems. The example presents the great effectiveness of the method that can shorten the length of calculation over hundreds times.
- Record ID
- CUTb722d2086e17490b9853ac75d9d8a8ba
- Publication categories
- ;
- Author
- Journal series
- Computational Mechanics, ISSN 0178-7675, e-ISSN 1432-0924
- Issue year
- 2018
- Vol
- 62
- No
- 1
- Pages
- 1-21
- Other elements of collation
- il. (w tym kolor.); Bibliografia (na s.) - 20-21; Bibliografia (liczba pozycji) - 51; Oznaczenie streszczenia - Abstr.; Data udostępnienia on-line - 2017-09-19; Numeracja w czasopiśmie - Vol. 62, Iss. 1
- Keywords in English
- discontinuous Galerkin method, orthogonal polynomials, high-order approximation, reference elements
- DOI
- DOI:10.1007/s00466-017-1479-z Opening in a new tab
- URL
- https://link.springer.com/article/10.1007/s00466-017-1479-z Opening in a new tab
- Language
- eng (en) English
- License
- Score (nominal)
- 45
- Additional fields
- Indeksowana w: Web of Science, Scopus
- Uniform Resource Identifier
- https://cris.pk.edu.pl/info/article/CUTb722d2086e17490b9853ac75d9d8a8ba/
- URN
urn:pkr-prod:CUTb722d2086e17490b9853ac75d9d8a8ba
* 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.