Direct Solvers for the Biharmonic Eigenvalue Problems Using Legendre Polynomials

We propose an efficient algorithm based on the Legendre–Galerkin approximations for the direct solution of the biharmonic eigenvalue problems with the boundary conditions of the clamped plate, the simply supported plate and the Cahn–Hilliard type. The key point to the efficiency of our algorithm is to construct appropriate basis functions which satisfying the corresponding boundary condition automatically and leading to linear systems with sparse matrices for the discrete variational formulations. In addition, the error estimate was driven by the minimax principle. Finally, the numerical results demonstrate the accuracy and the efficiency of this method.