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Discussion of Python Implementation Techniques for Discontinuous Galerkin Methods

Vestberg, Ulf LU (2023) In Bachelor's Theses in Mathematical Sciences NUMK11 20231
Mathematics (Faculty of Sciences)
Centre for Mathematical Sciences
Abstract
This paper discusses implementation techniques for integration methods within the Discontinuous Galerkin Methods. These methods are used to approximate solutions for differential equations. To do so, one must compute a polynomial, which is an approximation of the used function. To compute this approximation, orthogonal polynomials, chosen to be the Legendre polynomials are used, as well as integration. This is first demonstrated for one dimension. For two or three dimensions, full tensor product or sum-factorization is used. These implementations give the same approximation of the polynomial, but sum-factorization uses fewer computations than full tensor product. It is demonstrated how the the approximation is computed in Python and also... (More)
This paper discusses implementation techniques for integration methods within the Discontinuous Galerkin Methods. These methods are used to approximate solutions for differential equations. To do so, one must compute a polynomial, which is an approximation of the used function. To compute this approximation, orthogonal polynomials, chosen to be the Legendre polynomials are used, as well as integration. This is first demonstrated for one dimension. For two or three dimensions, full tensor product or sum-factorization is used. These implementations give the same approximation of the polynomial, but sum-factorization uses fewer computations than full tensor product. It is demonstrated how the the approximation is computed in Python and also that sum-factorization is faster to compute than the full tensor product based approach. (Less)
Please use this url to cite or link to this publication:
author
Vestberg, Ulf LU
supervisor
organization
course
NUMK11 20231
year
type
M2 - Bachelor Degree
subject
keywords
Best approximation, Orthogonal polynomials, Legendre polynomials, Integration, Full tensor product, Sum-factorization, Dune
publication/series
Bachelor's Theses in Mathematical Sciences
report number
LUNFNA-4047-2023
ISSN
1654-6229
other publication id
2023:K13
language
English
id
9123876
date added to LUP
2023-07-05 15:53:03
date last changed
2023-07-05 15:53:03
@misc{9123876,
  abstract     = {{This paper discusses implementation techniques for integration methods within the Discontinuous Galerkin Methods. These methods are used to approximate solutions for differential equations. To do so, one must compute a polynomial, which is an approximation of the used function. To compute this approximation, orthogonal polynomials, chosen to be the Legendre polynomials are used, as well as integration. This is first demonstrated for one dimension. For two or three dimensions, full tensor product or sum-factorization is used. These implementations give the same approximation of the polynomial, but sum-factorization uses fewer computations than full tensor product. It is demonstrated how the the approximation is computed in Python and also that sum-factorization is faster to compute than the full tensor product based approach.}},
  author       = {{Vestberg, Ulf}},
  issn         = {{1654-6229}},
  language     = {{eng}},
  note         = {{Student Paper}},
  series       = {{Bachelor's Theses in Mathematical Sciences}},
  title        = {{Discussion of Python Implementation Techniques for Discontinuous Galerkin Methods}},
  year         = {{2023}},
}