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. 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.

Nitsche 方法是一种行之有效的弱执行偏微分方程 (PDE) 边界条件的方法。它具有许多理想的特性，包括保持变分一致性，以及它产生对称、正定离散线性系统，并且不会过度病态。近年来，该方法在等几何分析、浸入法和接触力学等多个领域得到了普及。然而，得出基于尼采方法的公式可能是一个数学上艰巨的过程，尤其是对于高阶偏微分方程。幸运的是，在变分问题的背景下，推导在概念上是简单的。本文的目的是通过一系列教学示例来阐明该过​​程。首先，我们展示泊松方程的 Nitsche 方法的推导，以获得各个步骤的直观感受。接下来，我们提出抽象框架，然后重新审视泊松方程的推导，以使用该框架并增加数学严谨性。在此过程中，我们扩展推导以涵盖向量值设置。有了基本的秘诀，我们就会展示如何通过考虑向量值双调和方程和线性化基尔霍夫-洛夫板来处理高阶问题。最后，希望读者能够将尼采的方法应用于由变分原理引起的任何问题。