论文标题
构建Nitsche的变异问题方法
Constructing Nitsche's method for variational problems
论文作者
论文摘要
Nitsche的方法是弱执行部分微分方程（PDES）的弱执行边界条件的方法。它具有许多理想的属性，包括保留变分的一致性以及它产生对称的，正定的离散线性系统，这些系统未过度条件。近年来，该方法在许多领域中广受欢迎，包括等几何分析，浸入方法和接触力学。但是，基于Nitsche方法的公式可能是数学上艰巨的过程，尤其是对于高阶PDE的过程。幸运的是，在变化问题的背景下，该派生在概念上很简单。为了促进该过程，我们设计了一个抽象框架，用于在[J. Benzaken，J。A。Evans，S。McCormick和R. Tamstorf，Nitsche的线性Kirchhoff-love shells方法：配方，误差分析和验证，Comput。方法应用。机械。 Eng。，374（2021），p。 113544]。本文的目的是通过一系列教学示例阐明该过程。首先，我们展示了Nitsche的泊松方程方法的推导，以获得各个步骤的直觉。接下来，我们提出抽象框架，然后重新审视泊松方程的推导，以使用框架并添加数学严格。在此过程中，我们扩展了派生以涵盖矢量值设置。然后，我们配备基本食谱，然后通过考虑矢量值Biharmonic方程和线性化的Kirchhoff-love板来展示如何解决高阶问题。最后，希望读者能够将Nitsche的方法应用于各种原理引起的任何问题。
Nitsche's method is a well-established approach for weak enforcement of boundary conditions for partial differential equations (PDEs). It has many desirable properties, including the preservation of variational consistency and the fact that it yields symmetric, positive-definite discrete linear systems that are not overly ill-conditioned. In recent years, the method has gained in popularity in a number of areas, including isogeometric analysis, immersed methods, and contact mechanics. However, arriving at a formulation based on Nitsche's method can be a mathematically arduous process, especially for high-order PDEs. Fortunately, the derivation is conceptually straightforward in the context of variational problems. To facilitate the process, we devised an abstract framework for constructing Nitsche's method for these types of problems in [J. Benzaken, J. A. Evans, S. McCormick, and R. Tamstorf, Nitsche's method for linear Kirchhoff-Love shells: Formulation, error analysis, and verification, Comput. Methods Appl. Mech. Eng., 374 (2021), p. 113544]. The goal of this paper is to elucidate the process through a sequence of didactic examples. First, we show the derivation of Nitsche's method for Poisson's equation to gain an intuition for the various steps. Next, we present the abstract framework and then revisit the derivation for Poisson's equation to use the framework and add mathematical rigor. In the process, we extend our derivation to cover the vector-valued setting. Armed with a basic recipe, we then show how to handle a higher-order problem by considering the vector-valued biharmonic equation and the linearized Kirchhoff-Love plate. In the end, the hope is that the reader will be able to apply Nitsche's method to any problem that arises from variational principles.
