论文标题
在最小二乘设置的椭圆形PDE中，一种未固定的RBF-FD方法
An unfitted RBF-FD method in a least-squares setting for elliptic PDEs on complex geometries
论文作者
论文摘要
PDE的径向基函数生成有限差（RBF-FD）方法需要一组符合计算域$ω$的插值点。导致近似鲁棒性的要求之一是将插值点放置在$ω$的边界周围的局部均匀距离。但是，使用此类属性生成插值点是一个麻烦的问题。取而代之的是，可以将插值点扩展到边界上，并完全与$ω$的形状分离。在本文中，我们对最小二乘RBF-FD方法进行了修改，该方法允许将插值点放在封装$ω$的框中。这样，大大简化了2D和3D中复杂域上的节点放置。在复杂2D几何上求解椭圆模型PDE的数值实验表明我们的方法是强大的。此外，与经典的RBF-FD方法相比，它在近似误差和运行时与错误方面的性能更好。也可以在3D中使用我们的方法，我们通过在胸膜隔膜上提供溶液的收敛结果来指示。
Radial basis function generated finite difference (RBF-FD) methods for PDEs require a set of interpolation points which conform to the computational domain $Ω$. One of the requirements leading to approximation robustness is to place the interpolation points with a locally uniform distance around the boundary of $Ω$. However generating interpolation points with such properties is a cumbersome problem. Instead, the interpolation points can be extended over the boundary and as such completely decoupled from the shape of $Ω$. In this paper we present a modification to the least-squares RBF-FD method which allows the interpolation points to be placed in a box that encapsulates $Ω$. This way, the node placement over a complex domain in 2D and 3D is greatly simplified. Numerical experiments on solving an elliptic model PDE over complex 2D geometries show that our approach is robust. Furthermore it performs better in terms of the approximation error and the runtime vs. error compared with the classic RBF-FD methods. It is also possible to use our approach in 3D, which we indicate by providing convergence results of a solution over a thoracic diaphragm.
