Radial basis function generated finite difference (RBF-FD) methods for PDEs require a set of interpolation points which conform to the computational domain $\Omega$. One of the requirements leading to approximation robustness is to place the interpolation points with a locally uniform distance around the boundary of $\Omega$. 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 $\Omega$. 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 $\Omega$. 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.
翻译:辐射基函数生成了PDE的有限差异( RBF-FD) 方法。 导致近似稳健性的要求之一是在$\Omega$的边界周围放置具有当地统一距离的内插点。 但是, 产生这种特性的内插点是一个麻烦问题。 相反, 内插点可以扩展至边界范围, 从而完全脱离$\ Omega$的形状。 在本文中, 我们提出了一个符合计算域 $\ Omega$ 的内插点。 允许将内插点放置在一个包装$\ Omega$的盒子中。 这样, 2D 和 3D 的复杂域上的节点位置会大大简化。 在解决离子模型 PDE 和 复杂 2D 的地理模型时, 数字实验表明我们的方法是稳健的。 此外, 与典型的 RBFFF- FD 方法相比, 我们的近似误差和运行时间与运行时的误差也更好。 这样, 我们也可以使用3D 方法的趋同。