An efficient Fourier spectral method and error analysis for the fourth order problem with periodic boundary conditions and variable coefficients

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.