Based on the diagonalization technique and matrix decomposition technique, a new series of Legendre basis functions is constructed, which are simultaneously orthogonal in both \(L^2\) - and \(H^1\) -inner products, and lead to diagonal systems for second-order problems. Then we construct efficient spectral and space-time spectral methods for multidimensional problems using Legendre approximation in space and Legendre-Gauss collocation method in time, which can be implemented in a synchronous parallel fashion. Meanwhile, a new Legendre spectral element methods for solving high oscillation or steep gradient solutions problems are proposed, which reduce the non-zero entries of linear systems and computational cost. Numerical experiments exhibit the effectiveness and accuracy of the suggested approaches.
