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：本文研究了一个多边形域内带简支边界条件的六阶方程。我们提出了一个新的混合形式，将六阶问题分解为一组泊松方程。根据域的内角，可能需要额外的泊松问题来将解限制在正确的 Sobolev 空间中。此外，我们提出了一种 $C^0$ 有限元算法用于求解六阶问题，并提供了最优误差分析。我们还报告了数值结果以验证理论发现。