A Modification of the Fictitious Domain Method with Brinkman Penalization and a Boundary Condition for Pressure for the Nonlinear Navier–Stokes Model

Abstract

This paper investigates a modification of the fictitious domain method based on extension through lower-order coefficients, with a prescribed pressure boundary condition, for the nonlinear unsteady Navier–Stokes equations governing the motion of an incompressible homogeneous fluid in a bounded three-dimensional domain. A nonlinear Brinkman penalization term with parameter 0≤β<1, is introduced in the auxiliary domain. The existence and convergence of at least one global-in-time weak solution of the auxiliary problem are proved, whereas the existence, uniqueness, and convergence of a strong solution are established locally in time. For the strong solution, an estimate for the squared energy error was obtained with the exponents 16−6k16+12k−16β−3βk and 11−β, 00, the resulting convergence order is higher than the classical ε order in the norm. The mathematical role of the pressure condition on the outer boundary of the auxiliary domain and the influence of the parameter β are discussed.