Efficient Legendre-Laguerre spectral element methods for problems on unbounded domains with diagonalization technique

Based on the generalized matrix eigenvalue decomposition technique, two kinds of basis functions are 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 mixed Legendre-Laguerre spectral element methods for solving high oscillation or steep gradient solutions problems on unbounded domains, which reduce the non-zero entries of linear systems and computational cost. We also analyze the optimal convergence of numerical solutions and describe its numerical implementation. Numerical experiments exhibit the effectiveness and accuracy of the suggested approaches.