In this paper we study the biharmonic equation with Navier boundary conditions in a polygonal domain. In particular, we propose a method that effectively decouples the fourth-order problem as a system of Poisson equations. Our method differs from the naive mixed method that leads to two Poisson problems but only applies to convex domains; our decomposition involves a third Poisson equation to confine the solution in the correct function space, and therefore can be used in both convex and nonconvex domains. A $C^0$ finite element algorithm is in turn proposed to solve the resulting system. In addition, we derive optimal error estimates for the numerical solution on both quasi-uniform meshes and graded meshes. Numerical test results are presented to justify the theoretical findings.

中文翻译：

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多边形域内具有纳维边界条件的双调和问题的C0有限元法

在本文中，我们研究了多边形域中具有 Navier 边界条件的双调和方程。特别是，我们提出了一种有效地将四阶问题解耦为泊松方程系统的方法。我们的方法不同于导致两个泊松问题但仅适用于凸域的朴素混合方法；我们的分解涉及第三个泊松方程，以将解限制在正确的函数空间中，因此可以在凸域和非凸域中使用。进而提出了一个$C^0$ 有限元算法来求解所得系统。此外，我们得出了准均匀网格和分级网格上数值解的最优误差估计。给出了数值测试结果来证明理论发现的合理性。