Quadratic and higher order finite elements are interesting candidates for the numerical solution of PDE`s due to their improved approximation properties in comparison to linear/bilinear approaches. The linear systems that arise from the discretization of the underlying differential equation are very often solved by iterative solvers like CG-, BiCGStab, GMRES or others. Multigrid solvers are rarely used, which might be caused by the high effort that is associated with the appropriate numerical realization of smoothers and intergrid transfer operators.
In this talk we discuss the numerical analysis of the quadratic conforming finite element $Q_2$ in a multigrid solver. Numerical tests indicate that~-- if the problem is smooth enough and the correct grid transfer operator is provided~-- this element provides much better convergence rates than the use of linear/bilinear finite element spaces
like $Q_1$: If $m$ denotes the number of smoothing steps, the convergence rates behave like $O(/frac{1}{m2})$ in contrast to $O(/frac{1}{m})$ for first order FEM.