A $C^0$ finite element algorithm for the sixth order problem with simply supported boundary conditions

In this paper, we investigate a sixth order elliptic equation with the simply supported boundary conditions in a polygonal domain. We propose a new method that decouples the sixth order problem into a system of second order equations. Unlike the direct decomposition, which yields three Poisson problems but is restricted to polygonal domains with the largest interior angle no more than ${\pi}/{2}$, we rigorously analyze and construct extra Poisson problems to confine the solution into the same function space as that of the original sixth order problem. Consequently, the proposed method can be applied to general polygonal domains. In turn, we also present a $C^0$ finite element algorithm to discretize the new resulting system and establish optimal error estimates for the numerical solution on quasi-uniform meshes. Finally, numerical experiments are performed to validate the theoretical findings.