Two-scale Finite Element Discretizations for Semilinear Parabolic Equations

Fang Liu

doi:doi:10.11648/j.acm.20200906.12

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Two-scale Finite Element Discretizations for Semilinear Parabolic Equations

Fang Liu

Published in Applied and Computational Mathematics (Volume 9, Issue 6)

Received: 5 February 2020 Accepted: 25 September 2020 Published: 16 November 2020

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Abstract

In this paper, to reduce the computational cost of solving semilinear parabolic equations on a tensor product domain Ω⊂ℝ^d with d = 2 or 3, some two-scale finite element discretizations are proposed and analyzed. The time derivative in semilinear parabolic equations is approximated by the backward Euler finite difference scheme. The two-scale finite element method is designed for the space discretization. The idea of the two-scale finite element method is based on an understanding of a finite element solution to an elliptic problem on a tensor product domain. The high frequency parts of the finite element solution can be well captured on some univariate fine grids and the low frequency parts can be approximated on a coarse grid. Thus the two-scale finite element approximation is defined as a linear combination of some standard finite element approximations on some univariate fine grids and a coarse grid satisfying H = O (h^1/2), where h and H are the fine and coarse mesh widths, respectively. It is shown theoretically and numerically that the backward Euler two-scale finite element solution not only achieves the same order of accuracy in the H¹ (Ω) norm as the backward Euler standard finite element solution, but also reduces the number of degrees of freedom from O(h^-d×τ^-1) to O(h^-((d)+1)/2×τ^-1) where τ is the time step. Consequently the backward Euler two-scale finite element method for semilinear parabolic equations is more efficient than the backward Euler standard finite element method.

Published in	Applied and Computational Mathematics (Volume 9, Issue 6)
DOI	10.11648/j.acm.20200906.12
Page(s)	179-186
Creative Commons	This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.
Copyright	Copyright © The Author(s), 2020. Published by Science Publishing Group

Keywords

Two-scale, Finite Element, Combination, Semilinear Parabolic Equation

References

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Fang Liu. (2020). Two-scale Finite Element Discretizations for Semilinear Parabolic Equations. Applied and Computational Mathematics, 9(6), 179-186. https://doi.org/10.11648/j.acm.20200906.12

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ACS Style

Fang Liu. Two-scale Finite Element Discretizations for Semilinear Parabolic Equations. Appl. Comput. Math. 2020, 9(6), 179-186. doi: 10.11648/j.acm.20200906.12

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AMA Style

Fang Liu. Two-scale Finite Element Discretizations for Semilinear Parabolic Equations. Appl Comput Math. 2020;9(6):179-186. doi: 10.11648/j.acm.20200906.12

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@article{10.11648/j.acm.20200906.12,
  author = {Fang Liu},
  title = {Two-scale Finite Element Discretizations for Semilinear Parabolic Equations},
  journal = {Applied and Computational Mathematics},
  volume = {9},
  number = {6},
  pages = {179-186},
  doi = {10.11648/j.acm.20200906.12},
  url = {https://doi.org/10.11648/j.acm.20200906.12},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.acm.20200906.12},
  abstract = {In this paper, to reduce the computational cost of solving semilinear parabolic equations on a tensor product domain Ω⊂ℝd with d = 2 or 3, some two-scale finite element discretizations are proposed and analyzed. The time derivative in semilinear parabolic equations is approximated by the backward Euler finite difference scheme. The two-scale finite element method is designed for the space discretization. The idea of the two-scale finite element method is based on an understanding of a finite element solution to an elliptic problem on a tensor product domain. The high frequency parts of the finite element solution can be well captured on some univariate fine grids and the low frequency parts can be approximated on a coarse grid. Thus the two-scale finite element approximation is defined as a linear combination of some standard finite element approximations on some univariate fine grids and a coarse grid satisfying H = O (h1/2), where h and H are the fine and coarse mesh widths, respectively. It is shown theoretically and numerically that the backward Euler two-scale finite element solution not only achieves the same order of accuracy in the H1 (Ω) norm as the backward Euler standard finite element solution, but also reduces the number of degrees of freedom from O(h-d×τ-1) to O(h-((d)+1)/2×τ-1) where τ is the time step. Consequently the backward Euler two-scale finite element method for semilinear parabolic equations is more efficient than the backward Euler standard finite element method.},
 year = {2020}
}

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TY  - JOUR
T1  - Two-scale Finite Element Discretizations for Semilinear Parabolic Equations
AU  - Fang Liu
Y1  - 2020/11/16
PY  - 2020
N1  - https://doi.org/10.11648/j.acm.20200906.12
DO  - 10.11648/j.acm.20200906.12
T2  - Applied and Computational Mathematics
JF  - Applied and Computational Mathematics
JO  - Applied and Computational Mathematics
SP  - 179
EP  - 186
PB  - Science Publishing Group
SN  - 2328-5613
UR  - https://doi.org/10.11648/j.acm.20200906.12
AB  - In this paper, to reduce the computational cost of solving semilinear parabolic equations on a tensor product domain Ω⊂ℝd with d = 2 or 3, some two-scale finite element discretizations are proposed and analyzed. The time derivative in semilinear parabolic equations is approximated by the backward Euler finite difference scheme. The two-scale finite element method is designed for the space discretization. The idea of the two-scale finite element method is based on an understanding of a finite element solution to an elliptic problem on a tensor product domain. The high frequency parts of the finite element solution can be well captured on some univariate fine grids and the low frequency parts can be approximated on a coarse grid. Thus the two-scale finite element approximation is defined as a linear combination of some standard finite element approximations on some univariate fine grids and a coarse grid satisfying H = O (h1/2), where h and H are the fine and coarse mesh widths, respectively. It is shown theoretically and numerically that the backward Euler two-scale finite element solution not only achieves the same order of accuracy in the H1 (Ω) norm as the backward Euler standard finite element solution, but also reduces the number of degrees of freedom from O(h-d×τ-1) to O(h-((d)+1)/2×τ-1) where τ is the time step. Consequently the backward Euler two-scale finite element method for semilinear parabolic equations is more efficient than the backward Euler standard finite element method.
VL  - 9
IS  - 6
ER  -

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Author Information

Fang Liu

School of Statistics and Mathematics, Central University of Finance and Economics, Beijing, China

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Fang Liu. (2020). Two-scale Finite Element Discretizations for Semilinear Parabolic Equations. Applied and Computational Mathematics, 9(6), 179-186. https://doi.org/10.11648/j.acm.20200906.12

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ACS Style

Fang Liu. Two-scale Finite Element Discretizations for Semilinear Parabolic Equations. Appl. Comput. Math. 2020, 9(6), 179-186. doi: 10.11648/j.acm.20200906.12

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AMA Style

Fang Liu. Two-scale Finite Element Discretizations for Semilinear Parabolic Equations. Appl Comput Math. 2020;9(6):179-186. doi: 10.11648/j.acm.20200906.12

Copy | Download

@article{10.11648/j.acm.20200906.12,
  author = {Fang Liu},
  title = {Two-scale Finite Element Discretizations for Semilinear Parabolic Equations},
  journal = {Applied and Computational Mathematics},
  volume = {9},
  number = {6},
  pages = {179-186},
  doi = {10.11648/j.acm.20200906.12},
  url = {https://doi.org/10.11648/j.acm.20200906.12},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.acm.20200906.12},
  abstract = {In this paper, to reduce the computational cost of solving semilinear parabolic equations on a tensor product domain Ω⊂ℝd with d = 2 or 3, some two-scale finite element discretizations are proposed and analyzed. The time derivative in semilinear parabolic equations is approximated by the backward Euler finite difference scheme. The two-scale finite element method is designed for the space discretization. The idea of the two-scale finite element method is based on an understanding of a finite element solution to an elliptic problem on a tensor product domain. The high frequency parts of the finite element solution can be well captured on some univariate fine grids and the low frequency parts can be approximated on a coarse grid. Thus the two-scale finite element approximation is defined as a linear combination of some standard finite element approximations on some univariate fine grids and a coarse grid satisfying H = O (h1/2), where h and H are the fine and coarse mesh widths, respectively. It is shown theoretically and numerically that the backward Euler two-scale finite element solution not only achieves the same order of accuracy in the H1 (Ω) norm as the backward Euler standard finite element solution, but also reduces the number of degrees of freedom from O(h-d×τ-1) to O(h-((d)+1)/2×τ-1) where τ is the time step. Consequently the backward Euler two-scale finite element method for semilinear parabolic equations is more efficient than the backward Euler standard finite element method.},
 year = {2020}
}

Copy | Download

TY  - JOUR
T1  - Two-scale Finite Element Discretizations for Semilinear Parabolic Equations
AU  - Fang Liu
Y1  - 2020/11/16
PY  - 2020
N1  - https://doi.org/10.11648/j.acm.20200906.12
DO  - 10.11648/j.acm.20200906.12
T2  - Applied and Computational Mathematics
JF  - Applied and Computational Mathematics
JO  - Applied and Computational Mathematics
SP  - 179
EP  - 186
PB  - Science Publishing Group
SN  - 2328-5613
UR  - https://doi.org/10.11648/j.acm.20200906.12
AB  - In this paper, to reduce the computational cost of solving semilinear parabolic equations on a tensor product domain Ω⊂ℝd with d = 2 or 3, some two-scale finite element discretizations are proposed and analyzed. The time derivative in semilinear parabolic equations is approximated by the backward Euler finite difference scheme. The two-scale finite element method is designed for the space discretization. The idea of the two-scale finite element method is based on an understanding of a finite element solution to an elliptic problem on a tensor product domain. The high frequency parts of the finite element solution can be well captured on some univariate fine grids and the low frequency parts can be approximated on a coarse grid. Thus the two-scale finite element approximation is defined as a linear combination of some standard finite element approximations on some univariate fine grids and a coarse grid satisfying H = O (h1/2), where h and H are the fine and coarse mesh widths, respectively. It is shown theoretically and numerically that the backward Euler two-scale finite element solution not only achieves the same order of accuracy in the H1 (Ω) norm as the backward Euler standard finite element solution, but also reduces the number of degrees of freedom from O(h-d×τ-1) to O(h-((d)+1)/2×τ-1) where τ is the time step. Consequently the backward Euler two-scale finite element method for semilinear parabolic equations is more efficient than the backward Euler standard finite element method.
VL  - 9
IS  - 6
ER  -

Copy | Download