An Efficient Legendre–Galerkin Approximation for Fourth-Order Elliptic Problems with SSP Boundary Conditions and Variable Coefficients

Hui Zhang, Xingrong Yang, Jiulin Jin, Xu Zhang and Jun Zhang ()
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Hui Zhang: School of Information, Guizhou University of Finance and Economics, Guiyang 550025, China
Xingrong Yang: School of Management, Hefei University of Technology, Hefei 230009, China
Jiulin Jin: College of Mathematics and Information Science, Guiyang University, Guiyang 550005, China
Xu Zhang: Financial Department, Guizhou University of Finance and Economics, Guiyang 550025, China
Jun Zhang: Computational Mathematics Research Center, Guizhou University of Finance and Economics, Guiyang 550025, China

Mathematics, 2023, vol. 11, issue 10, 1-16

Abstract: 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.

Keywords: fourth-order problem; SSP boundary condition; Legendre polynomial approximation; error analysis (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2023
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