Partial Decomposition of a Domain Containing Thin Tubes for Solving the Diffusion Equation

Abstract

In a domain containing thin cylindrical tubes, we consider the diffusion equation with the Neumann boundary condition on the lateral surface of the tubes. The problem is reduced to a problem of hybrid dimension so that the reduced problem has the original dimension outside the tubes, but is reduced to the one-dimensional diffusion equation inside the tubes. The docking of models of different dimensions is carried out according to the method of asymptotic partial decompositions of domains. We estimate the difference between the solutions to the initial problem and the problem of different dimension. Bibliography: 14 titles. Illustrations: 3 figures.

On a Stability of Non-Stationary Discrete Schemes with Respect to Interpolation Errors

“…In order to show that the constructed three-level discrete scheme on this special nonuniform time grid ω t is stable and additional grid points really reduce the global error of the discrete solution, we give also the errors Z 3 of the classical three-level discrete scheme (14) when the time grid is uniform and it has N discrete points. The results of the computational experiments are presented in Table 1, where Z 1 is the error for the discrete solution of the scheme ( 12) and ( 13), Z 2 is the error for the discrete solution of the classical finite-difference scheme (14) on non-uniform time grids, and Z 3 is the error of the solution of the symmetrical scheme (7) when the time grid is uniform in [0, 1] and it has N points.…”

“…Errors ∥Z 1 ∥ ∞ and experimental convergence rates ρ(τ) at T = 1 for the discrete solution of the scheme ( 12) and ( 13) and errors ∥Z 2 ∥ ∞ and experimental convergence rates ρ(τ) for the discrete solution of the finite-difference scheme (14) for a sequence of time steps τ. ∥Z 3 ∥ ∞ is the error of the discrete solution of the the symmetrical scheme (7) when the time grid is uniform and it has N points.…”

“…where N is the selected number of time points in the first subinterval. The results of the computational experiments are given in Table 2, where Z 1 is the error for the discrete solution of the scheme ( 18), Z 2 is the error for the discrete solution of the classical finite-difference scheme (14) on non-uniform time grids, and ρ 1,2 (τ) are experimental convergence rates.…”

“…The general convergence framework states that the stability and approximation properties guarantee the convergence of discrete solutions [1,2]. The deep, broad, and constructive theories of approximation and stability are developed, and they cover various important topics dealing with non-smooth data [3][4][5], weak solutions [6][7][8], energy and maximum principle stability estimates [1,2,9,10], ill-posed and inverse problems [11,12], and nonlocal mathematical models including fractional derivatives [13][14][15][16].…”