Efficient and Accurate Legendre Spectral Element Methods for One-Dimensional Higher Order Problems

Abstract

Efficient and accurate Legendre spectral element methods for solving one-dimensional higher order differential equations with high oscillatory or steep gradient solutions are proposed. Some Sobolev orthogonal/biorthogonal basis functions corresponding to each subinterval are constructed, which reduce the non-zero entries of linear systems and computational cost. Numerical experiments exhibit the effectiveness and accuracy of the suggested approaches.