Some estimates for the viscoelastic incompressible Kelvin-Voigt medium

Abstract

In this paper, we consider the application of the method of fictitious domains to a viscoelastic incompressible medium based on the Kelvin-Voigt model. Application of the method of fictitious domains allows solving the original problem in regions with complex geometric configuration. This makes it easier to automate the construction of a consistent difference mesh, and to solve the problem in areas of standard shape. Estimates for the proximity of the auxiliary problem’s solution are obtained. The auxiliary problem is constructed by the method of fictitious domains. These estimates refer to the solution of the original problem. The original problem describes a viscoelastic incompressible medium. Convergence follows from the estimates of the proximity of the solutions of the original and auxiliary problems. Further, on the basis of the method of fictitious domains, two-sided estimates on a small parameter for the difference between the solution of the original problem and the solution of the auxiliary problem constructed by the method of fictitious domains are obtained. Moreover, the solution to the auxiliary problem is expanded as a series in powers of the small parameter. This is possible because that solution is represented as a functional series that converges absolutely in the original domain.