A Spectral Method for Fourth-Order Mixed Inhomogeneous Boundary Value Problem in Three Dimensions

In this paper, we investigate spectral method for fourth- order mixed inhomogeneous boundary value problems in three dimensions. Some results on the three-dimensional Legendre approximation for fourth- order problem in Jacobi weighted Sobolev space are established, which improve and generalize the existing results, and play an important role in numerical solutions of partial differential equations. We also develop a lifting technique for fourth-order problems, with which we could handle mixed inhomogeneous boundary conditions easily. As examples of applications, spectral schemes are provided for three model problems with mixed inhomogeneous boundary conditions. The spectral accuracy in space of proposed algorithms are proved. Efficient implementations are presented. Numerical results demonstrate their high accuracy, and confirm the theoretical analysis well.