On a method for solving three-dimensional elliptic equation of a general type (Q2729345)

The authors present the so-called iterated interpolation method for the numerical solution of an elliptic equation of the form NEWLINE\[NEWLINE \frac{\partial T}{\partial t}=\sum_{m=1}^3\frac{\partial}{\partial x_m}\left(v_m\frac{\partial T} {\partial x_m}\right) + w\sum_{_{\substack{ m = 1\\ k\neq m}}}^3 \frac{\partial^2 T} {\partial x_m\partial x_k} + \sum_{m=1}^3u_m\frac{\partial T}{\partial x_m} - c_4T + f = 0,\quad w = \text{const},\;c_4 > 0, NEWLINE\]NEWLINE in the parallelepiped \(R = \{(x_1,x_2,x_3): 0 < x_m < d_m,\;m = 1,2,3\}\), under the following boundary value conditions: NEWLINE\[NEWLINE T|_{x_1 = d_1} = p_1(d_1,x_2,x_3),\quad T|_{x_2 = d_2} = p_2(x_1,d_2,x_3), \quad T|_{x_3 = d_3} = p_3(x_1,x_2,d_3),NEWLINE\]NEWLINE NEWLINE\[NEWLINET|_{x_1 = 0} = s_1(x_2,x_3),\quad T|_{x_2 = 0} = s_2(x_1,x_3), \quad T|_{x_3 = 0} = s_3(x_1,x_2).NEWLINE\]NEWLINE To solve the problem, the authors consider a difference analog of the above-mentioned problem. Finite-difference schemes are constructed on the basis of a generalized iterated interpolation method. To prove convergence of the iterative process, the relaxation method is applied.NEWLINENEWLINENEWLINEIn applications, schemes of this type are used for numerical solution of the Poisson equation. A theoretical evaluation of the convergence rate is presented for the three-dimensional Poisson equation with constant transport coefficients.