$C^1$-Conforming Quadrilateral Spectral Element Method for Fourth-Order Equations

Abstract

This paper is devoted to Professor Benyu Guo’s open question on the $C^1$-conforming quadrilateral spectral element method for fourth-order equations which has been endeavored for years. Starting with generalized Jacobi polynomials on the reference square, we construct the $C^1$-conforming basis functions using the bilinear mapping from the reference square onto each quadrilateral element which fall into three categories—interior modes, edge modes, and vertex modes. In contrast to the triangular element, compulsively compensatory requirements on the global $C^1$-continuity should be imposed for edge and vertex mode basis functions such that their normal derivatives on each common edge are reduced from rational functions to polynomials, which depend on only parameters of the common edge. It is amazing that the $C^1$-conforming basis functions on each quadrilateral element contain polynomials in primitive variables, the completeness is then guaranteed and further confirmed by the numerical results on the Petrov–Galerkin spectral method for the non-homogeneous boundary value problem of fourth-order equations on an arbitrary quadrilateral. Finally, a $C^1$-conforming quadrilateral spectral element method is proposed for the biharmonic eigenvalue problem, and numerical experiments demonstrate the effectiveness and efficiency of our spectral element method.