On the Use of Dual Variational Principles for the Estimation of Error in Approximate Solutions of Diffusion Problems

Two-sided estimates for thermal resistance of an inhomogeneous solid body

“…This MM can be reduced to a dual variational problem of stationary heat transfer [14][15][16][17], including two different functionals, that reach one and the same extreme value (minimum and maximum), and determine the total thermal resistance of the solid body under consideration [18]. An approximate solu tion to the problem of finding extreme values of the functionals allows us to obtain a two sided estimate of the thermal resistance of interest.…”

“…T 2 (x 2 ) = T 2 + , which are allowable tem perature distributions for functional (5) in the areas with the thermal conductivity coefficients λ 1 and λ 2 , respectively ( Fig. 1), that contain the unknown values G 1 = and G 2 = Then for the temperature difference in the contact plane x 2 = x 1 tanβ, we obtain (14) where ΔT = T 2 -T 1 . As a result, functional (5) takes the form Replacing Eq.…”

“…As a result, functional (5) takes the form Replacing Eq. (14) and calculating the integral yields (15) Since functional (5) reaches the minimum on true temperature distributions, the values G 1 and G 2 can be obtained using the necessary conditions of minimum = 0 and = 0 for func tional (15). This leads to a system of two linear alge braic equations with respect to G 1 and G 2 , the solution of which has the form (16) where Z = + + b = Bi = = For functional (10), the simplest allowable vector of the heat flux density corresponds to the value q = const for the component of this vector in the positive direction of the Ox 2 axis.…”

Dual variational principles for an elliptic partial differential equation

“…Let us introduce the following real functions: 1) a tj (x), a(x) are bounded piecewise continuous functions in Q with jumps on F f and such that for x e Q° they satisfy (1) a 0 (x) = a } {x) , i, j = 1, ..., N ,…”

“…The boundary values of functions belonging to W 1,2 (Q) will be understood in the sense of traces and we shall often write u instead of Tu for the values on the boundary.…”