Volumetric variational principles for a class of partial differential equations defined on surfaces and curves
Volumetric variational principles for a class of partial differential equations defined on surfaces and curves
复制标题
曲面和曲线上定义的一类偏微分方程的体积变分原理
DOI：
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发表时间：
2017
影响因子：
1.2
通讯作者：
R. Tsai
中科院分区：
文献类型：
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作者：
J. Chu;R. Tsai
在本文中，我们提出了简单的数值算法的偏微分方程（PDE）定义在封闭的，光滑的表面（或曲线）。特别是，我们认为偏微分方程，源于变分原理定义的表面上，这些包括拉普拉斯-贝尔特拉米方程和表面波方程。该方法是系统地制定扩展的变分积分，并推导出扩展问题的欧拉-拉格朗日方程，包括边界条件，可以很容易地离散均匀笛卡尔网格或自适应网格。在我们的方法中，表面被定义的距离函数或最近的点映射隐式。由于这样的扩展是不唯一的，我们研究如何一类简单的扩展可以影响所产生的偏微分方程。特别是，我们减少了表面偏微分方程的模型问题上定义的周期性带和相应的边界条件，并使用经典的傅立叶和拉普拉斯变换方法来研究所产生的问题的适定性。对于椭圆和抛物型问题，我们的边界封闭主要产生稳定的算法来解决非线性曲面偏微分方程。对于双曲型问题，建议的边界封闭一般是不稳定的，但不稳定性可以很容易地通过添加高阶正则化项或通过定期但不经常地“重新初始化”计算的解决方案来控制。对每种典型的曲面偏微分方程给出了数值算例。
In this paper, we propose simple numerical algorithms for partial differential equations (PDEs) defined on closed, smooth surfaces (or curves). In particular, we consider PDEs that originate from variational principles defined on the surfaces; these include Laplace–Beltrami equations and surface wave equations. The approach is to systematically formulate extensions of the variational integrals and derive the Euler–Lagrange equations of the extended problem, including the boundary conditions that can be easily discretized on uniform Cartesian grids or adaptive meshes. In our approach, the surfaces are defined implicitly by the distance functions or by the closest point mapping. As such extensions are not unique, we investigate how a class of simple extensions can influence the resulting PDEs. In particular, we reduce the surface PDEs to model problems defined on a periodic strip and the corresponding boundary conditions and use classical Fourier and Laplace transform methods to study the well-posedness of the resulting problems. For elliptic and parabolic problems, our boundary closure mostly yields stable algorithms to solve nonlinear surface PDEs. For hyperbolic problems, the proposed boundary closure is unstable in general, but the instability can be easily controlled by either adding a higher-order regularization term or by periodically but infrequently “reinitializing” the computed solutions. Some numerical examples for each representative surface PDEs are presented.
影响因子：
14.2
作者：
Dziuk, Gerhard;Elliott, Charles M.
通讯作者：
Elliott, Charles M.