In this paper, we present a novel approach to the approximate solution of elliptic partial differential equations on compact submanifolds of \(\mathbb {R}^{d}\), particularly compact surfaces and the surface equation \({\Delta }_{\mathbb {M}} u - \lambda u=f\). In the course of this, we reconsider differential operators on such submanifolds to deduce suitable penalty based functionals. These functionals are based on the residual of the equation in an integral representation, extended by a penalty on the first-order normal derivative. The general framework we develop is accompanied by error analysis and exemplified by numerical examples employing tensor product B-splines.

中文翻译：

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嵌入子流形上偏微分方程的环境残罚近似

在本文中，我们为\（\ mathbb {R} ^ {d} \）的紧子流形，特别是紧致曲面和表面方程\（{\ Delta} _ {\ mathbb {M}} u-\ lambda u = f \）。在此过程中，我们将重新考虑这些子流形上的微分算子，以得出合适的基于惩罚的泛函。这些函数基于积分表示式中的方程残差，并通过一阶正态导数的罚分来扩展。我们开发的通用框架伴随着误差分析，并通过使用张量积B样条的数值示例进行了举例说明。