An Efficient Legendre–Galerkin Approximation for Fourth-Order Elliptic Problems with SSP Boundary Conditions and Variable Coefficients
Under simply supported plate (SSP) boundary conditions, a numerical method based on the higher-order Legendre polynomial approximation was studied and developed for fourth-order problems with variable coefficients. We first divide the SSP boundary conditions into two types, namely, forced boundary c...
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MDPI AG
2023-05-01
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| Series: | Mathematics |
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| Online Access: | https://www.mdpi.com/2227-7390/11/10/2236 |
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| author | Hui Zhang Xingrong Yang Jiulin Jin Xu Zhang Jun Zhang |
| author_facet | Hui Zhang Xingrong Yang Jiulin Jin Xu Zhang Jun Zhang |
| author_sort | Hui Zhang |
| collection | DOAJ |
| description | Under simply supported plate (SSP) boundary conditions, a numerical method based on the higher-order Legendre polynomial approximation was studied and developed for fourth-order problems with variable coefficients. We first divide the SSP boundary conditions into two types, namely, forced boundary conditions and natural boundary conditions. According to the forced boundary conditions, an appropriate Sobolev space is defined, and a variational formulation and a discrete scheme associated with the original problem are established. Then, the existence and uniqueness of this weak solution and approximate solution are proved. By using the Céa lemma and the tensor Jacobian polynomial approximation, we further obtain the error estimation for the numerical solutions. In addition, we use the orthogonality of Legendre polynomials to construct a set of effective basis functions and derive the equivalent tensor product linear system associated with the discrete scheme, respectively. Finally, some numerical tests were carried out to validate our algorithm and theoretical analysis. |
| first_indexed | 2024-03-11T03:32:01Z |
| format | Article |
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| issn | 2227-7390 |
| language | English |
| last_indexed | 2024-03-11T03:32:01Z |
| publishDate | 2023-05-01 |
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| series | Mathematics |
| spelling | doaj.art-28fc422603cb4ca1aede820a391de6512023-11-18T02:18:03ZengMDPI AGMathematics2227-73902023-05-011110223610.3390/math11102236An Efficient Legendre–Galerkin Approximation for Fourth-Order Elliptic Problems with SSP Boundary Conditions and Variable CoefficientsHui Zhang0Xingrong Yang1Jiulin Jin2Xu Zhang3Jun Zhang4School of Information, Guizhou University of Finance and Economics, Guiyang 550025, ChinaSchool of Management, Hefei University of Technology, Hefei 230009, ChinaCollege of Mathematics and Information Science, Guiyang University, Guiyang 550005, ChinaFinancial Department, Guizhou University of Finance and Economics, Guiyang 550025, ChinaComputational Mathematics Research Center, Guizhou University of Finance and Economics, Guiyang 550025, ChinaUnder simply supported plate (SSP) boundary conditions, a numerical method based on the higher-order Legendre polynomial approximation was studied and developed for fourth-order problems with variable coefficients. We first divide the SSP boundary conditions into two types, namely, forced boundary conditions and natural boundary conditions. According to the forced boundary conditions, an appropriate Sobolev space is defined, and a variational formulation and a discrete scheme associated with the original problem are established. Then, the existence and uniqueness of this weak solution and approximate solution are proved. By using the Céa lemma and the tensor Jacobian polynomial approximation, we further obtain the error estimation for the numerical solutions. In addition, we use the orthogonality of Legendre polynomials to construct a set of effective basis functions and derive the equivalent tensor product linear system associated with the discrete scheme, respectively. Finally, some numerical tests were carried out to validate our algorithm and theoretical analysis.https://www.mdpi.com/2227-7390/11/10/2236fourth-order problemSSP boundary conditionLegendre polynomial approximationerror analysis |
| spellingShingle | Hui Zhang Xingrong Yang Jiulin Jin Xu Zhang Jun Zhang An Efficient Legendre–Galerkin Approximation for Fourth-Order Elliptic Problems with SSP Boundary Conditions and Variable Coefficients Mathematics fourth-order problem SSP boundary condition Legendre polynomial approximation error analysis |
| title | An Efficient Legendre–Galerkin Approximation for Fourth-Order Elliptic Problems with SSP Boundary Conditions and Variable Coefficients |
| title_full | An Efficient Legendre–Galerkin Approximation for Fourth-Order Elliptic Problems with SSP Boundary Conditions and Variable Coefficients |
| title_fullStr | An Efficient Legendre–Galerkin Approximation for Fourth-Order Elliptic Problems with SSP Boundary Conditions and Variable Coefficients |
| title_full_unstemmed | An Efficient Legendre–Galerkin Approximation for Fourth-Order Elliptic Problems with SSP Boundary Conditions and Variable Coefficients |
| title_short | An Efficient Legendre–Galerkin Approximation for Fourth-Order Elliptic Problems with SSP Boundary Conditions and Variable Coefficients |
| title_sort | efficient legendre galerkin approximation for fourth order elliptic problems with ssp boundary conditions and variable coefficients |
| topic | fourth-order problem SSP boundary condition Legendre polynomial approximation error analysis |
| url | https://www.mdpi.com/2227-7390/11/10/2236 |
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