论文标题
切割一类非线性椭圆PDE的有限元误差估计值
Cut finite element error estimates for a class of nonlinear elliptic PDEs
论文作者
论文摘要
虚构的域方法是由许多暴露于巨大拓扑变化或变形的复杂域中的应用程序的动机，将感兴趣的实际域视为嵌入在固定的笛卡尔背景中。这通常是通过其边界通过级别函数的几何参数化来实现的。在本说明中，对带有切割元素的未固定数值方案的先验分析超出了线性问题的领域。更确切地说，我们考虑通过切割有限元方法使用多项式非线性的$-ΔU +f_1（u）的半线性椭圆边界值问题的离散化。使用Nitsche型方法执行边界条件。为了确保独立于边界相对于网格的边界位置的稳定性和误差估计，用附加边界区域幽灵惩罚项增强了配方。这些术语对与切割元素相关的面部正常梯度的跳跃起作用。得出了A-Priori误差估计，而数值示例说明了该方法的实现并验证了理论发现。
Motivated by many applications in complex domains with boundaries exposed to large topological changes or deformations, fictitious domain methods regard the actual domain of interest as being embedded in a fixed Cartesian background. This is usually achieved via a geometric parameterization of its boundary via level-set functions. In this note, the a priori analysis of unfitted numerical schemes with cut elements is extended beyond the realm of linear problems. More precisely, we consider the discretization of semilinear elliptic boundary value problems of the form $- Δu +f_1(u)=f_2$ with polynomial nonlinearity via the cut finite element method. Boundary conditions are enforced, using a Nitsche-type approach. To ensure stability and error estimates that are independent of the position of the boundary with respect to the mesh, the formulations are augmented with additional boundary zone ghost penalty terms. These terms act on the jumps of the normal gradients at faces associated with cut elements. A-priori error estimates are derived, while numerical examples illustrate the implementation of the method and validate the theoretical findings.
