A Legendre spectral‐Galerkin method for fourth‐order problems in cylindrical regions

Abstract A spectral‐Galerkin method based on Legendre‐Fourier approximation for fourth‐order problems in cylindrical regions is studied in this paper. By the cylindrical coordinate transformation, a three‐dimensional fourth‐order problem in a cylindrical region is transformed into a sequence of decoupled fourth‐order problems with two dimensions and the corresponding pole conditions are also derived. With appropriately constructed weighted Sobolev space, a weak form is established. Based on this weak form, a spectral‐Galerkin discretization scheme is proposed and its error is rigorously analyzed by defining a new class of projection operators. Then, a set of efficient basis functions are used to write the discrete scheme as the linear systems with a sparse matrix based on tensor product. Numerical examples are presented to show the efficiency and high‐accuracy of the developed method. Finally, an application of the proposed method to the fourth‐order Steklov problem and the corresponding numerical experiments once again confirm the efficiency and spectral accuracy of the method.