Structured Spectral Element Methods Based On The Non-standard Basis Functions And Their Applications

Finite difference method,finite element method and spectral method are three im-portant methods in solving partial differential equations.It is noteworthy that spectral method based on orthogonal polynomials which enjoy high accuracy,that is,smoother the underlying solution is,better the approximation is.In this paper,we propose a new kind of spectral and spectral element methods for high order problems.We first introduce a family of new basis functions by matrix decomposition technique for the fourth order problems with constant coefficients ho-mogeneous boundary conditions.Next,based on the new basis functions,we propose a new space-time spectral method,and the designed algorithms not only ensure the high precision but also save the calculation time.We prove its spectral accuracy both in space and time.The numerical results also show the correctness of theoretical anal-ysis.Finally,by using the advantages of the new proposed basis functions,we show the solving scheme for fourth-order problems in one-dimensional and two-dimensional.The results demonstrate its efficiency of spectral element methods.