In this paper we consider a class of fictitious domain finite element methods known from the literature. These methods use standard finite element spaces on a fixed unfitted triangulation combined with the Nitsche technique and a ghost penalty stabilization. As a model problem we consider the application of such a method to the Poisson equation. We introduce and analyze a new class of preconditioners that is based on a subspace decomposition approach. The finite element space is split into two subspaces, where one subspace is spanned by all nodal basis functions corresponding to nodes on the boundary of the fictitious domain and the other space is spanned by all remaining nodal basis functions. We will show that this splitting is stable, uniformly in the discretization parameter and in the location of the problem boundary in the triangulation. We also prove that the Galerkin discretization in the first subspace leads to a uniformly well-conditioned matrix and that the Galerkin discretization in the second subspace is uniformly equivalent to a standard finite element discretization of a Poisson equation on the fictitious domain with homogeneous Dirichlet boundary conditions. Results of numerical experiments that illustrate optimality of such a preconditioner are included.
翻译:在本文中,我们考虑从文献中知道的一类虚域有限元素方法。 这种方法使用标准有限元素空间, 固定不适三角格, 加上尼采技术和幽灵惩罚稳定。 作为示范问题, 我们考虑将这种方法应用于 Poisson 方程式。 我们引入和分析基于子空间分解法的新型先决条件。 有限元素空间分为两个子空间, 其中一个子空间由与虚域边界上的节点相对应的所有节点基函数跨越, 而另一个空间则由所有其余的节点基函数跨越。 我们将显示这种分裂是稳定的, 在离散参数和问题边界位置在三角中是一致的。 我们还证明, 第一个子空间的加勒金离散导致一个条件一致的矩阵, 第二个子空间的加勒金离散功能与虚构域的普瓦森方程式的标准有限要素离异化一致, 与同一的dirichlet边界条件相同。 数字实验的结果说明了这种先决条件的最佳性。