论文标题
对通用域中泊松问题的移动边界方法的分析
Analysis of the Shifted Boundary Method for the Poisson Problem in General Domains
论文作者
论文摘要
移位边界方法（SBM）是在更广泛的未嵌入/嵌入/沉浸式方法中，是边界价值问题的近似域方法。事实证明，它在处理复杂几何形状的问题方面非常有效，从泊松到达西，从纳维尔 - 斯托克斯到弹性及其他。 SBM的关键特征是在应用dirichlet边界条件的位置 - 从真实边界到替代边界的位置 - 以及对边界条件值的适当修改（再次，{\ it Shift shift}），以减少一致性误差。在本文中，我们对平滑和非平滑域中的方法进行了声音分析，突出了确切和替代边界对收敛速率之间的几何形状和距离的影响。在不失去一般性的情况下，我们将托里奇特边界条件的泊松问题视为模型，我们首先详细介绍了获得替代和真实边界之间关键转移的程序。接下来，我们为离散问题的适当性和稳定性提供了足够的条件。彻底分析了通过转移边界条件而产生的一致性误差的行为，对于平滑边界和边界的边界和边缘和边缘的边界。事实证明，收敛速率在能源规范中是最佳的，并且在$ l^2 $ norm中得到了进一步增强。
The shifted boundary method (SBM) is an approximate domain method for boundary value problems, in the broader class of unfitted/embedded/immersed methods. It has proven to be quite efficient in handling problems with complex geometries, ranging from Poisson to Darcy, from Navier-Stokes to elasticity and beyond. The key feature of the SBM is a {\it shift} in the location where Dirichlet boundary conditions are applied - from the true to a surrogate boundary - and an appropriate modification (again, a {\it shift}) of the value of the boundary conditions, in order to reduce the consistency error. In this paper we provide a sound analysis of the method in smooth and non-smooth domains, highlighting the influence of geometry and distance between exact and surrogate boundaries upon the convergence rate. Without loss of generality, we consider the Poisson problem with Dirichlet boundary conditions as a model and we first detail a procedure to obtain the crucial shifting between the surrogate and the true boundaries. Next, we give a sufficient condition for the well-posedness and stability of the discrete problem. The behavior of the consistency error arising from shifting the boundary conditions is thoroughly analyzed, for smooth boundaries and for boundaries with corners and edges. The convergence rate is proven to be optimal in the energy norm, and is further enhanced in the $L^2$-norm.
