We propose in this paper an efficient algorithm based on the Fourier spectral-Galerkin approximation for the fourth-order elliptic equation with periodic boundary conditions and variable coefficients. First, by using the Lax-Milgram theorem, we prove the existence and uniqueness of weak solution and its approximate solution. Then we define a high-dimensional $L^2$ projection operator and prove its approximation properties. Combined with Céa lemma, we further prove the error estimate of the approximate solution. In addition, from the Fourier basis function expansion and the properties of the tensor, we establish the equivalent matrix form based on tensor product for the discrete scheme. Finally, some numerical experiments are carried out to demonstrate the efficiency of the algorithm and correctness of the theoretical analysis.

In this paper, we developed and studied an efficient Legendre spectral method for fourth order eigenvalue problems with the boundary conditions of a simply supported plate. Initially, a new variational formulation based on a hybrid format and its discrete variational form were established. We then employed the spectral theory of complete continuous operators to establish the prior error estimates of the approximate solutions. By integrating approximation results of some orthogonal projection operators in weighted Sobolev spaces, we further gave the error estimation for the approximating eigenvalues and eigenfunctions. In addition, we developed an effective set of basis functions by utilizing the orthogonal properties of Legendre polynomials, and subsequently derived the matrix eigenvalue system of the discrete variational form for both two-dimensional and three-dimensional cases, based on a tensor product. Finally, numerical examples were provided to demonstrate the exponential convergence and efficiency of the algorithm.