Method of Fictitious Domains and Homotopy as a New Alternative to Multidimensional Partial Differential Equations in Domains of Any Shape

Abstract

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).