A Modified Brinkman Penalization Fictitious Domain Method for the Unsteady Navier‐Stokes Equations

This paper investigates a modification of the fictitious domain method with continuation in the lower‐order coefficients for the unsteady Navier‐Stokes equations governing the motion of an incompressible homogeneous fluid in a bounded 2D or 3D domain. The modification enables a solution‐dependent choice of the critical parameter. Global‐in‐time existence and convergence of a weak solution to the auxiliary problem are proved, and local‐in‐time existence and convergence of a unique strong solution are established. For the strong solution, a new higher‐order convergence rate estimate in the penalization parameter is obtained. The introduced framework allows us to apply a pointwise divergence‐free finite element method as a discretization technique, leading to a strongly mass conservative discrete fictitious domain method. A numerical example illustrates the performance of the method.