论文标题
复杂几何形状中椭圆形PDE的局部基础差异潜在方法
Local-basis Difference Potentials Method for elliptic PDEs in complex geometry
论文作者
论文摘要
在差异电势框架中，我们为复杂几何形状中的椭圆形部分微分方程开发了有效且高阶的准确有限差异方法。开发方案的主要新颖性是使用在近边界网格点定义的局部基础函数。局部函数的使用允许对（i）明确和隐式定义的几何形状进行统一的数值处理； （ii）形状更复杂的几何形状，例如有角落，多连接域等的几何形状； （iii）不同类型的边界条件。这种几何灵活的方法是使用全局基础函数的经典差异方法补充的，尤其是在需要大量的全局基础函数以解决边界或难以获得最佳全局基础函数的情况下。基于FFT的快速泊松求解器用于标准的中心有限差模板，而不论精确的精确顺序如何。理论上和数字上都概述了最大规范中差异电位收敛的证据。
We develop efficient and high-order accurate finite difference methods for elliptic partial differential equations in complex geometry in the Difference Potentials framework. The main novelty of the developed schemes is the use of local basis functions defined at near-boundary grid points. The use of local basis functions allow unified numerical treatment of (i) explicitly and implicitly defined geometry; (ii) geometry of more complicated shapes, such as those with corners, multi-connected domain, etc; and (iii) different types of boundary conditions. This geometrically flexible approach is complementary to the classical difference potentials method using global basis functions, especially in the case where a large number of global basis functions are needed to resolve the boundary, or where the optimal global basis functions are difficult to obtain. Fast Poisson solvers based on FFT are employed for standard centered finite difference stencils regardless of the designed order of accuracy. Proofs of convergence of difference potentials in maximum norm are outlined both theoretically and numerically.
