Method of Fictitious Domains and Homotopy as a New Alternative to Multidimensional Partial Differential Equations in Dom
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METHOD OF FICTITIOUS DOMAINS AND HOMOTOPY AS A NEW ALTERNATIVE TO MULTIDIMENSIONAL PARTIAL DIFFERENTIAL EQUATIONS IN DOMAINS OF ANY SHAPE I. P. Gavrilyuk1,2 and V. L. Makarov3
UDC 517.9; 519.63
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 O(N log N ) operations).
Introduction In the process of solution of multidimensional partial differential equations in domains of any shape, one may encounter certain difficulties caused by the geometries of these domains. In the solution of problems of this kind by the finite-element method, it is necessary, first, to triangulate the domain, which complicates the entire algorithm of solution. Thus, if we use the 2D Delaunay algorithm for the generation of a grid with N inner nodes, then we get an additional number of computing operations proportional to O(N log N ). The indicated geometric difficulties stimulated the development of the method of fictitious domains in which the original domain of any shape is immersed in a parallelepiped that can be easily covered with a multidimensional rectangular grid, which can be easily (and even trivially) made finer (whenever necessary). This approach enables one to use regular grids and, hence, special algorithms of solution (solvers) and predeterminators in problems with complicated geometry [1]. The method of fictitious domains, together with the finite-element method, enables one, in particular, to significantly increase the degree of automation of the process of programming, which significantly simplifies the transfer from one applied problem to another. These factors of technological efficiency of the proposed method, in combination with the finite-element method, prove to be very important for the creation of the applied software packages [1]. In [2], the method of fictitious domains was applied for the first time to a problem of the following form posed in an arbitrary domain ⌦ :
Lu(x) ⌘ −
✓ ◆ n X @ @u aij (x) + c(x)u = f (x), @xi @xj
i,j=1
u(x) = 0,
x = (x1 , . . . , xn ) 2 ⌦,
(1)
x 2 @⌦,
1
Duale Hochschule Gera-Eisenach, Gera, Germany; e-mail: [email protected]. Corresponding author. 3 Institute of Mathematics, Ukrainian National Academy of Sciences, Kyiv, Ukraine; e-mail: [email protected]. 2
Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 72, No. 2, pp. 191–208, February, 2020. Original article submitted October 20, 2019. 0041-5995/20/7202–0211
© 2020
Springer Science+Business Media, LLC
211
I. P. G AVRILYUK
212
AND
V. L. M AKAROV
with a condition c(x) ≥ 0,
aij (x) = aji (x), and the following condition of ellipti
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