Graddiv-conforming spectral element method for fourth-order div problems

Abstract

This paper introduces a novel numerical method to solve fourth-order div problems using $\text{graddiv}$-conforming spectral elements on cuboidal meshes. We start by determining the continuity requirements for $\text{graddiv}$-conforming spectral elements, followed by constructing these elements using generalized Jacobi polynomials and the Piola transformation. The resulting basis functions exhibit a hierarchical structure, making them easily extendable to higher orders. We apply these $\text{graddiv}$-conforming spectral elements to solve the fourth-order div problem and present numerical examples to verify both the efficiency and effectiveness of the method.