Multigrid Methods For Mortar-type Q₁～(rot) And Q₁～(rot)/Qo Elements

In this thesis, we discuss multigrid algorithms for mortar-type rotated Q₁ element for second order elliptic problems and mortar-type Q₁^rot/Q₀ element for the incompressible Stokes problem. For second order elliptic problems, we propose an intergrid transfer operator for the nonested mortar element spaces. It is proven that the W-cycle multigrid method is optimal, i.e., the convergence rate is independent of the mesh size and mesh level. Meanwhile, a variable V-cycle multigrid algorithm is presented, which provides a precondi-tioner with a uniformly bounded condition number. Numerical experiments demonstrate that the optimal convergence property for the W-cycle algorithm holds with any number of smoothing steps. For the incompressible Stokes problem, we firstly propose mortar element method for Q₁^rot/Q₀ element for it, and get the optimal error estimate by proving the inf-sup condition of the discrete saddle point problem. Then a similar intergrid transfer operator is given for the spaces of velocity, and the W-cycle multigrid method is presented for solving the algebraic equations. Finally the optimal convergence rate for the velocity of the multigrid is proven, and numerical experiments are given to confirm our theoretical results.