论文标题
Darcy流的差异有限元方法的差异
A divergence preserving cut finite element method for Darcy flow
论文作者
论文摘要
我们研究了基于混合有限元对$ \ textbf {rt} _k \ times q_k $，$ k \ geq 0 $的cut cut cut darcy接口问题的有限元离散化。这里$ q_k $是不连续的多项式函数的空间，该功能较少或等于$ k $，而$ \ textbf {rt} $是raviart-thomas Space。我们表明，标准的幽灵惩罚稳定稳定，通常以较弱的剪切有限元方法的形式添加，以稳定和控制线性系统矩阵的条件数量，从而破坏了所考虑的元素对的无差异属性。因此，我们提出了压力的新稳定项，并表明我们恢复了发散的最佳近似，而不会失去对线性系统矩阵的条件数量的控制。我们证明，具有新稳定项的方法具有螺线管速度场的无差异近似值。我们基于$ \ textbf {rt} _k \ times q_k $，$ k \ geq 0 $的提议的未限制元素离散化的先验错误估算。此外，通过将网格分解为宏观元素并仅在宏观元素的内部边缘应用幽灵惩罚条款，就非常有限地应用稳定，仅在需要时应用。具有元素对的数值实验$ \ textbf {rt} _0 \ times q_0 $，$ \ textbf {rt} _1 _1 \ times q_1 $，和$ \ textbf {bdm} _1 _1 \ times q_0 q_0 q_0 $ 1）近似速度和压力的最佳收敛速率； 2）适合的线性系统，其中系统矩阵量表的条件数与适合有限元离散化的条件数量一样； 3）近似差异的最佳收敛速率与螺线管速度场的无差差近似值。这三个属性都独立于界面相对于计算网格的定位方式。
We study cut finite element discretizations of a Darcy interface problem based on the mixed finite element pairs $\textbf{RT}_k\times Q_k$, $k\geq 0$. Here $Q_k$ is the space of discontinuous polynomial functions of degree less or equal to $k$ and $\textbf{RT}$ is the Raviart-Thomas space. We show that the standard ghost penalty stabilization, often added in the weak forms of cut finite element methods for stability and control of the condition number of the linear system matrix, destroys the divergence-free property of the considered element pairs. Therefore, we propose new stabilization terms for the pressure and show that we recover the optimal approximation of the divergence without losing control of the condition number of the linear system matrix. We prove that the method with the new stabilization term has pointwise divergence-free approximations of solenoidal velocity fields. We derive a priori error estimates for the proposed unfitted finite element discretization based on $\textbf{RT}_k\times Q_k$, $k\geq 0$. In addition, by decomposing the mesh into macro-elements and applying ghost penalty terms only on interior edges of macro-elements, stabilization is applied very restrictively and only where needed. Numerical experiments with element pairs $\textbf{RT}_0\times Q_0$, $\textbf{RT}_1\times Q_1$, and $\textbf{BDM}_1\times Q_0$ (where $\textbf{BDM}$ is the Brezzi-Douglas-Marini space) indicate that we have 1) optimal rates of convergence of the approximate velocity and pressure; 2) well-posed linear systems where the condition number of the system matrix scales as it does for fitted finite element discretizations; 3) optimal rates of convergence of the approximate divergence with pointwise divergence-free approximations of solenoidal velocity fields. All three properties hold independently of how the interface is positioned relative to the computational mesh.
