Legendre-Galerkin spectral-element method for the biharmonic equations and its applications

A spectral-element method based on the Legendre-Galerkin approximation is presented to solve the two-dimensional biharmonic equations. Rigorous error analysis is carried out to establish the convergence of the method. By constructing appropriate basis functions which satisfy the boundary conditions of the differential equations, the discrete variational formulation is reduced to linear system with sparse and symmetric matrices, which can be efficiently solved by a fast Schur-complement method. Accuracy test is provided to confirm the convergence rate of the theoretical results. Finally, the proposed method is applied to calculate the displacement of an elastic plate under a uniform applied load and stream function of zero Reynolds number flow in a driven cavity, the results compare well with established results.