Mortar Finite Element Method For Stokes Problem

The mortar finite element method introduced by Bernardi-Maday-Patera(1993) [3] is particulary useful while designing complex structural components which may have been modelled by different analysts, and these analysts often work on several individual subdomains. In such case, it is often too cumbersome, or even infeasible, to coordinate the mesh over separate components such that they are conforming at interfaces. Hence, one needs to seek help of nonconforming (mortar) techniques. Moreover since discretization in some localized regions need to be increased, which contribute the most to the error in any problem, these techniques are useful for applications in the regions (such as around corner and so on). To our knowledge, the matching constraints at the interfaces of different subdomains can be expressed in terms of Lagrange multipliers which produce good approximation of normal derivatives of the real solution across mortar edges, this technique is a generalization of mortar finite element method.It is well known that there exist many applications in viscous incompressible flows. For example, the design of hydraulic turbines, the process of plastics or metals molten, and so on. In this thesis, we only consider the simplest possible incompressible flow problem, i.e., the steady-state Stokes problem.In the first part we discuss mortar-type P₁^NC — P₀ approximation for Stokes problem. The well-known inf-sup condition is proved and the optimal error estimate is obtained. Meanwhile, a multigrid algorithm is proposed for solving this discrete problem, and we prove that the W-cycle multigrid is optimal, i.e., the convergence rate is independent of the mesh size and mesh level. Finally, numerical experiments are presented to confirm our theoretical results.In the second part we propose a mortar element method with La-grange multipliers for Stokes problem, i.e., the matching constraintsof velocity on mortar edges are expressed in terms of Lagrange multipliers. By proving two inf-sup conditions, we obtain optimal error estimates for velocity and pressure, and we get good approximation of normal derivatives of the velocity across the interfaces.Throughout this thesis, we denote by "C" a universal constant which is independent of the mesh size and level, but whose value can differ from place to place.