In this paper, we study the sixth order equation with the simply supported boundary conditions in a polygonal domain. We propose a new mixed formulation that decomposes the sixth order problem into a system of Poisson equations. Depending on the interior angles of the domain, additional Poisson problems may be needed to confine the solution to the correct Sobolev space. In addition, we propose a $C^0$ finite element algorithm for the sixth order problem and provide the optimal error analysis. Numerical results are reported to verify the theoretical findings.

翻译：本文研究多边形区域上具有简支边界条件的六阶方程。我们提出了一种新的混合形式，将六阶问题分解为泊松方程组。根据区域的内角情况，可能需要附加的泊松问题以将解约束在正确的Sobolev空间中。此外，我们针对该六阶问题提出了一类$C^0$有限元算法，并给出了最优误差分析。数值结果验证了理论结论。