Error estimates of spectral Legendre-Galerkin methods for the fourth-order equation in one dimension

We employ spectral Legendre-Galerkin and mixed Legendre-Galerkin approximations to solve the first bi-harmonic equation in one dimension, respectively. By orthogonal properties of Legendre polynomials, we obtain an explicit a posteriori error indicator for spectral Legendre-Galerkin methods. Furthermore, in virtue of an auxiliary variable, we present spectral mixed Legendre-Galerkin methods and study the a priori estimate and a posteriori error indicator. Especially, these indicators only depend on the expansions of the right-hand item. Numerical examples are presented to verify Off theoretical analysis. (C) 2015 Elsevier Inc. All rights reserved.