The ideas of the method of fictitious domains and homotopy are combined with an aim to reduce the solution of boundary-value problems for multidimensional partial differential equations (PDE) in domains of any shape to an exponentially convergent sequence of PDE in a parallelepiped (or, in the 2D case, in a rectangle). This enables us to decrease the required computer time due to the elimination of the necessity of triangulation of the domain by a grid with N inner nodes (thus, the Delaunay algorithm in the 2D case requires $$ \mathcal{O} $$ (N log N) operations).

中文翻译：

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虚拟域和同伦方法作为任何形状域中多维偏微分方程的新替代方法

虚拟域和同伦方法的思想相结合，旨在将任何形状域中多维偏微分方程 (PDE) 的边值问题的解简化为平行六面体中 PDE 的指数收敛序列（或，在二维情况下，在矩形中）。由于消除了通过具有 N 个内部节点的网格对域进行三角剖分的必要性，这使我们能够减少所需的计算机时间（因此，二维情况下的 Delaunay 算法需要 $$ \mathcal{O} $$ (N记录 N) 个操作）。