@article{V.L. Makarov_Gavrilyuk_2020, title={The fictitious domain method and homotopy as a new alternative for multidimensional partial differential equations in domains of any shape}, volume={72}, url={http://umj.imath.kiev.ua/index.php/umj/article/view/1101}, abstractNote={<p>UDC 517.9; 519.63</p> <p>The ideas of the fictitious domain method 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 PDEs in a parallelepiped (in a rectangle, in the 2D case). <br>This allows us to reduce the computational costs due to the elimination of the necessity of triangulation of the domain by a grid with $N$ inner nodes (e.g., the Delaunay algorithm in the 2D case requires ${\mathcal {O }(N \log{N})$ operations).</p>}, number={2}, journal={Ukrainsâ€™kyi Matematychnyi Zhurnal}, author={V.L. Makarov and Gavrilyuk, I. P.}, year={2020}, month={Feb.}, pages={191-208} }