We extend the application of Legendre-Galerkin algorithms for sixth-order elliptic problems with\r\nconstant coefficients to sixth-order elliptic equations with variable polynomial coefficients. The\r\ncomplexities of the algorithm are O(N) operations for a one-dimensional domain with N - 5\r\nunknowns. An efficient and accurate direct solution for algorithms based on the Legendre-\r\nGalerkin approximations developed for the two-dimensional sixth-order elliptic equations with\r\nvariable coefficients relies upon a tensor product process. The proposed Legendre-Galerkin\r\nmethod for solving variable coefficients problem is more efficient than pseudospectral method.\r\nNumerical examples are considered aiming to demonstrate the validity and applicability of the\r\nproposed techniques.
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