A Modified Brinkman-Type Fictitious-Domain Finite Element Method for Incompressible Flow with Rigid Obstacles

Moving-boundary problems for incompressible flows often require repeated mesh generation when body-fitted finite element methods are used. This may become costly for flows with moving rigid obstacles, since the numerical solution from the previous time level also has to be transferred to the new finite element space. In this paper, we study a fictitious-domain finite element method for the two-dimensional incompressible Navier–Stokes equations with rigid obstacles. The aim is to analyze a penalized formulation on a fixed background domain and to compare it with a classical body-fitted finite element approach. The flow problem is extended to a larger fixed domain, and the effect of the solid domain is imposed by a nonlinear penalty term depending on the parameters ε and β. A fully discrete finite element scheme is constructed, and stability and convergence estimates are obtained for the fixed-obstacle auxiliary penalized problem. The moving obstacle case is examined numerically. Several numerical tests are presented. A manufactured auxiliary problem is used to check the spatial convergence of the method, the influence of the penalty parameter, and the behavior of the solution inside the fictitious domain. We also consider two moving obstacle examples: a rotating disk moving near a no-slip wall and two rotating disks moving along symmetric trajectories. The fictitious-domain solutions are compared with body-fitted finite element solutions, and the main flow patterns are reproduced without remeshing. The results indicate that the fictitious-domain finite element method is a practical alternative for moving obstacle flow problems. It is not universally faster than body-fitted methods, but it becomes advantageous when the geometry changes in time, since the fixed background mesh avoids repeated remeshing and the transfer of the previous solution between nonmatching meshes.