A note on optimal multigrid convergence for higher-order FEM

Quadratic and even higher order finite elements are interesting candidates for the numerical solution of partial differential equations (PDEs) due to their improved approximation properties in comparison to linear approaches. The systems of equations that arise from the discretisation of the underlying (elliptic) PDEs are often solved by iterative solvers like preconditioned Krylow-space methods, while multigrid solvers are still rarely used – which might be caused by the high effort that is associated with the numerical realisation of smoothing and intergrid transfer operators. In this note, we discuss the numerical analysis of quadratic conforming finite elements in a multigrid solver. Numerical tests indicate that – if the ‘correct’ grid transfer is used – quadratic elements provide much better (asymptotic) convergence rates than linear finite element spaces: If m denotes the number of smoothing steps, the convergence rates behave asymptotically like O( 1 m2 ) in contrast to O( 1 m ) for linear FEM. The corresponding proof is explained in this note.