We formulate a well-posedness and approximation theory for a class of generalised saddle point problems with a specific form of constraints. In this way we develop an approach to a class of fourth order elliptic partial differential equations with point constraints using the idea of splitting into coupled second order equations. An approach is formulated using a penalty method to impose the constraints. Our main motivation is to treat certain fourth order equations involving the biharmonic operator and point Dirichlet constraints for example arising in the modelling of biomembranes on curved and flat surfaces but the approach may be applied more generally. The theory for well-posedness and approximation is presented in an abstract setting. Several examples are described together with some numerical experiments.

中文翻译：

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具有点约束的一类四阶偏微分方程的二阶分裂

我们为一类具有特定约束形式的广义鞍点问题制定了适定性和近似理论。通过这种方式，我们利用分裂成耦合二阶方程的思想，开发了一类具有点约束的四阶椭圆偏微分方程的方法。使用惩罚方法来制定一种方法来施加约束。我们的主要动机是处理某些涉及双调和算子和点狄利克雷约束的四阶方程，例如在曲面和平面上的生物膜建模中出现的，但该方法可能更广泛地应用。适定性和近似的理论是在一个抽象的环境中提出的。几个例子连同一些数值实验一起被描述。