New space–time spectral and structured spectral element methods for high order problems

We propose new space–time spectral and structured spectral element methods for high order problems. By matrix decomposition, we introduce a family of new basis functions which leads to a diagonal system for the fourth order problems with constant coefficients. Based on new basis functions in space a...
Ausführliche Beschreibung

We propose new space–time spectral and structured spectral element methods for high order problems. By matrix decomposition, we introduce a family of new basis functions which leads to a diagonal system for the fourth order problems with constant coefficients. Based on new basis functions in space and a dual-Petrov–Galerkin formulation in time, we propose a new space–time spectral method for a time-dependent problem, and proved their spectral accuracy both in space and time. Numerical results demonstrate their high efficiency and coincide well with theoretical analysis. Further, to ultimately promote the numerical performance and efficiency, we exploit the idea of locally simultaneous diagonalization to provide new structured spectral element methods for high-order problems. Besides, to increase the flexibility, a C 1 -conforming rectangular spectral method in analogy to Argyris or Bell triangular elements are proposed for fourth-order equations, which serves as a preparative work towards conforming quadrilateral spectral elements for high-order equations. Ausführliche Beschreibung