论文标题
$ c^0 $有限元法，用于多键式边界条件的双旋转问题
A $C^0$ finite element method for the biharmonic problem with Dirichlet boundary conditions in a polygonal domain
论文作者
论文摘要
在本文中，我们研究了多边形结构域中具有差异的边界条件的Biharmonic方程。特别是，我们提出了一种有效地将四阶问题解散到两个毒药方程和一个stokes方程的系统的方法，或一个stokes方程和一个泊松方程的系统。结果表明，每个系统的解决方案等效于凸和非凸多边形域上的原始四阶问题。另外提出了两种有限元算法来求解解耦系统。此外，我们还显示了Sobolev空间和加权Sobolev空间中每个解耦系统中解决方案的规律性，并得出了在准均匀网格和分级网格上的数值溶液的最佳误差估计值。提出了数值测试结果以证明理论发现合理。
In this paper, we study the biharmonic equation with Dirichlet boundary conditions in a polygonal domain. In particular, we propose a method that effectively decouples the fourth-order problem into a system of two Poison equations and one Stokes equation, or a system of one Stokes equation and one Poisson equation. It is shown that the solution of each system is equivalent to that of the original fourth-order problem on both convex and non-convex polygonal domains. Two finite element algorithms are in turn proposed to solve the decoupled systems. In addition, we show the regularity of the solutions in each decoupled system in both the Sobolev space and the weighted Sobolev space, and we derive the optimal error estimates for the numerical solutions on both quasi-uniform meshes and graded meshes. Numerical test results are presented to justify the theoretical findings.
