论文标题
不合格的有限元在三个维度
Nonconforming finite element Stokes complexes in three dimensions
论文作者
论文摘要
从构象的拉格朗日元素开始，以三个维度的stokes方程式以三个维度构建了两个不合格的有限元stokes sTokes复合物。通过结合不合格的有限元Stokes复合物和插值操作员，也显示了交换图。下级$ \ boldsymbol h（\ textrm {grad} \ textrm {curl}）$ - 不合格的有限元仅具有$ 14 $的自由度，其基本函数是根据barycentric coordinates明确给出的。 $ \ boldsymbol h（\ textrm {grad} \ textrm {curl}）$ - 不合格元素被应用于解决Quad-curl问题，并得出了最佳收敛性。通过不合格的有限元stokes络合物，Quad-Curl问题的混合有限元方法将其分解为Maxwell方程的两种混合方法，而不合格的$ P_1 $ - $ P_0 $ - $ P_0 $元素方法用于Stokes方程，基于该快速求解器的讨论。提供数值结果以验证理论收敛速率。
Two nonconforming finite element Stokes complexes starting from the conforming Lagrange element and ending with the nonconforming $P_1$-$P_0$ element for the Stokes equation in three dimensions are constructed. And commutative diagrams are also shown by combining nonconforming finite element Stokes complexes and interpolation operators. The lower order $\boldsymbol H(\textrm{grad}\textrm{curl})$-nonconforming finite element only has $14$ degrees of freedom, whose basis functions are explicitly given in terms of the barycentric coordinates. The $\boldsymbol H(\textrm{grad}\textrm{curl})$-nonconforming elements are applied to solve the quad-curl problem, and optimal convergence is derived. By the nonconforming finite element Stokes complexes, the mixed finite element methods of the quad-curl problem are decoupled into two mixed methods of the Maxwell equation and the nonconforming $P_1$-$P_0$ element method for the Stokes equation, based on which a fast solver is discussed. Numerical results are provided to verify the theoretical convergence rates.
