On Multigrid Convergence for Quadratic Finite Elements

Quadratic and higher order finite elements are interesting candidates for the numerical solution of (elliptic) partial differential equations (PDEs) due to their improved approximation properties in comparison to linear approaches. While the systems of equations that arise from the discretisation of the underlying PDEs are often solved by iterative schemes like preconditioned Krylow-space methods, multigrid solvers are still rarely used due the higher effort that is associated with the realization of appropriate smoothing and intergrid transfer operators. However, numerical tests indicate that quadratic FEM can provide even better convergence rates than linear finite elements: If m denotes the number of smoothing steps, the convergence rates behave asymptotically like O( 1 m 2 ) in contrast to O( 1 m ) for linear FEM. We prove this new convergence result for quadratic conforming finite elements in a multigrid solver.