论文标题
Laplace-Beltrami操作员的局部径向基函数方法
A local radial basis function method for the Laplace-Beltrami operator
论文作者
论文摘要
我们介绍了一种新的本地网格，用于在嵌入$ \ r^3 $中的co-Dimension One的平滑表面上近似Laplace-Beltrami操作员的近似。该方法的一个关键要素是它不需要表面的明确表达，可以简单地由一组散射的节点定义。它不需要表面正常向量的表达式和表面的曲率都不需要使用纸张中衍生的公式近似的表面。另一个优点是它是一种局部方法，因此，近似Laplace-Beltrami操作员的矩阵稀疏，可以转化为良好的可扩展性属性。研究了该方法的收敛，准确性和其他计算特征。通过求解表面上的两个反应扩散部分微分方程来显示性能；图案形成的图灵模型和Schaeffer的电气心组织行为模型。
We introduce a new local meshfree method for the approximation of the Laplace-Beltrami operator on a smooth surface of co-dimension one embedded in $\R^3$. A key element of this method is that it does not need an explicit expression of the surface, which can be simply defined by a set of scattered nodes. It does not require expressions for the surface normal vectors and for the curvature of the surface neither, which are approximated using formulas derived in the paper. An additional advantage is that it is a local method and, hence, the matrix that approximates the Laplace-Beltrami operator is sparse, which translates into good scalability properties. The convergence, accuracy and other computational characteristics of the method are studied numerically. The performance is shown by solving two reaction-diffusion partial differential equations on surfaces; the Turing model for pattern formation, and the Schaeffer's model for electrical cardiac tissue behavior.
