论文标题
一种新的$ ϕ $ -FEM方法，用于自然边界条件的问题
A new $ϕ$-FEM approach for problems with natural boundary conditions
论文作者
论文摘要
我们提出了一种称为$ ϕ $ -FEM的新的有限元方法，用于使用简单的计算网格使用自然（Neumann或Robin）边界条件求解数值椭圆形的部分微分方程，而不是适用于物理域的边界。边界数据使用级别集合函数考虑在内，这是处理复杂或不断发展的域的流行工具。我们的方法属于虚拟域方法（或浸入边界方法）的家族，并且接近CutFem/XFEM类型的最新方法。与后者相反，$ ϕ $ -FEM不需要在切割网格元素或实际边界上进行任何非标准数值集成，同时确保具有任何程度的有限元素的最佳收敛订单，并提供相当良好的离散问题。在第一个版本的$ ϕ $ -FEM中，仅考虑了必需的（dirichlet）边界条件。在这里，为了应对自然的边界条件，我们将主要解决方案的梯度作为辅助变量引入。这仅在由边界切割的网状细胞上完成，因此数值系统的大小仅略有增加。从理论上讲，我们证明了我们方案的最佳收敛性，并且与离散的问题条件结合，而与网格切割无关。数值实验证实了这些结果。
We present a new finite element method, called $ϕ$-FEM, to solve numerically elliptic partial differential equations with natural (Neumann or Robin) boundary conditions using simple computational grids, not fitted to the boundary of the physical domain. The boundary data are taken into account using a level-set function, which is a popular tool to deal with complicated or evolving domains. Our approach belongs to the family of fictitious domain methods (or immersed boundary methods) and is close to recent methods of cutFEM/XFEM type. Contrary to the latter, $ϕ$-FEM does not need any non-standard numerical integration on cut mesh elements or on the actual boundary, while assuring the optimal convergence orders with finite elements of any degree and providing reasonably well conditioned discrete problems. In the first version of $ϕ$-FEM, only essential (Dirichlet) boundary conditions was considered. Here, to deal with natural boundary conditions, we introduce the gradient of the primary solution as an auxiliary variable. This is done only on the mesh cells cut by the boundary, so that the size of the numerical system is only slightly increased. We prove theoretically the optimal convergence of our scheme and a bound on the discrete problem conditioning, independent of the mesh cuts. The numerical experiments confirm these results.
