Cascadic Multigrid Methods For Mortar-type Q₁～(rot) And Q₁～(rot)/Q₀ Elements

In this thesis, we discuss cascadic multigrid algorithms for mortar-type rotated Q₁ element for the second order elliptic problem and mortar-type Q₁^rot/Q₀ element for the incompressible Stokes problem. For the second order elliptic problem, we propose an intergrid transfer operator for the nonested mortar element spaces. It is proved that the cascadic conjugate gradinet method is optimal, i.e., the convergence rate is independent of the mesh size and mesh level. Meanwhile, the cascadic multigrid method with traditional iteration is nearly optimal. Numerical results confirm our theoretical analysis. For the incompressible Stokes problem, we first give a similar intergrid transfer operator for the nonested mortar element spaces and present the cascadic conjugate gradinet method for solving the algebraic equations. Then the optimal convergence rate of the cascadic multigrid for the velocity is proven.