An efficient Fourier spectral method and error analysis for the fourth order problem with periodic boundary conditions and variable coefficients
We propose in this paper an efficient algorithm based on the Fourier spectral-Galerkin approximation for the fourth-order elliptic equation with periodic boundary conditions and variable coefficients. First, by using the Lax-Milgram theorem, we prove the existence and uniqueness of weak solution and...
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| Format: | Article |
| Language: | English |
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AIMS Press
2023-02-01
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| Series: | AIMS Mathematics |
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| Online Access: | https://www.aimspress.com/article/doi/10.3934/math.2023484?viewType=HTML |
| _version_ | 1811158418092720128 |
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| author | Tingting Jiang Jiantao Jiang Jing An |
| author_facet | Tingting Jiang Jiantao Jiang Jing An |
| author_sort | Tingting Jiang |
| collection | DOAJ |
| description | We propose in this paper an efficient algorithm based on the Fourier spectral-Galerkin approximation for the fourth-order elliptic equation with periodic boundary conditions and variable coefficients. First, by using the Lax-Milgram theorem, we prove the existence and uniqueness of weak solution and its approximate solution. Then we define a high-dimensional $ L^2 $ projection operator and prove its approximation properties. Combined with Céa lemma, we further prove the error estimate of the approximate solution. In addition, from the Fourier basis function expansion and the properties of the tensor, we establish the equivalent matrix form based on tensor product for the discrete scheme. Finally, some numerical experiments are carried out to demonstrate the efficiency of the algorithm and correctness of the theoretical analysis. |
| first_indexed | 2024-04-10T05:24:19Z |
| format | Article |
| id | doaj.art-0827ba0b236e4ee48a1be4e5054785fe |
| institution | Directory Open Access Journal |
| issn | 2473-6988 |
| language | English |
| last_indexed | 2024-04-10T05:24:19Z |
| publishDate | 2023-02-01 |
| publisher | AIMS Press |
| record_format | Article |
| series | AIMS Mathematics |
| spelling | doaj.art-0827ba0b236e4ee48a1be4e5054785fe2023-03-08T01:14:58ZengAIMS PressAIMS Mathematics2473-69882023-02-01849585960110.3934/math.2023484An efficient Fourier spectral method and error analysis for the fourth order problem with periodic boundary conditions and variable coefficientsTingting Jiang0Jiantao Jiang 1Jing An2School of Mathematical Sciences, Guizhou Normal University, Guiyang 550025, ChinaSchool of Mathematical Sciences, Guizhou Normal University, Guiyang 550025, ChinaSchool of Mathematical Sciences, Guizhou Normal University, Guiyang 550025, ChinaWe propose in this paper an efficient algorithm based on the Fourier spectral-Galerkin approximation for the fourth-order elliptic equation with periodic boundary conditions and variable coefficients. First, by using the Lax-Milgram theorem, we prove the existence and uniqueness of weak solution and its approximate solution. Then we define a high-dimensional $ L^2 $ projection operator and prove its approximation properties. Combined with Céa lemma, we further prove the error estimate of the approximate solution. In addition, from the Fourier basis function expansion and the properties of the tensor, we establish the equivalent matrix form based on tensor product for the discrete scheme. Finally, some numerical experiments are carried out to demonstrate the efficiency of the algorithm and correctness of the theoretical analysis.https://www.aimspress.com/article/doi/10.3934/math.2023484?viewType=HTMLfourth-order problemsperiodic boundary conditionsvariable coefficientsfourier spectral methoderror estimation |
| spellingShingle | Tingting Jiang Jiantao Jiang Jing An An efficient Fourier spectral method and error analysis for the fourth order problem with periodic boundary conditions and variable coefficients AIMS Mathematics fourth-order problems periodic boundary conditions variable coefficients fourier spectral method error estimation |
| title | An efficient Fourier spectral method and error analysis for the fourth order problem with periodic boundary conditions and variable coefficients |
| title_full | An efficient Fourier spectral method and error analysis for the fourth order problem with periodic boundary conditions and variable coefficients |
| title_fullStr | An efficient Fourier spectral method and error analysis for the fourth order problem with periodic boundary conditions and variable coefficients |
| title_full_unstemmed | An efficient Fourier spectral method and error analysis for the fourth order problem with periodic boundary conditions and variable coefficients |
| title_short | An efficient Fourier spectral method and error analysis for the fourth order problem with periodic boundary conditions and variable coefficients |
| title_sort | efficient fourier spectral method and error analysis for the fourth order problem with periodic boundary conditions and variable coefficients |
| topic | fourth-order problems periodic boundary conditions variable coefficients fourier spectral method error estimation |
| url | https://www.aimspress.com/article/doi/10.3934/math.2023484?viewType=HTML |
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