Approaching of the solution of a static compressible medium to the solution of an incompressible medium

Abstract

Approaching of the solution of a static compressible medium to the solution of an incompressible medium A well-known analogy of the flow of viscous incompressible fluid and incompressible elastic medium. According to this analogy, the solution of the equations of the elasticity theory with the Poisson's ratio ν = 0, 5 and for any fixed shear modulus µ can be interpreted as a motion of a viscous incompressible fluid with viscosity µ. Thus, we can consider the usual static linear elasticity task with Hooke's law at λ → ∞, as a mathematical model of approaching to incompressible medium. In this paper, we obtained the asymptotic λ → ∞. Estimation of the proximity of the solution of an elastic static problem with Hooke's law to the solution of incompressible medium (Stokes problem). The final estimate allows to use well-known difference schemes and algorithms for an elastic compressible medium to solve incompressible medium. In this paper, an estimate of the proximity of the solutions of these problems is proved at λ → ∞, i.e. u→u H λ→∞ λ div u→−p λ→∞ σ→σ H λ→∞ . To substantiate this fact in [1-3], various methods for the first boundary value problem were investigated. For the static problem of the theory of elasticity, there is currently a whole series of papers devoted to numerical implementation using difference schemes. In paper [4], the estimate O(λ −α ) where k = 1 2 was obtained, in the proposed paper the estimate O(λ −1 ), and in further work we will show that this estimate is best possible in order.