On Multigrid Convergence for Quadratic Finite Elements MichaelK¨oster ∗ Stefan Turek † January 12, 2006 Abstract Quadratic and higher order finite elements are interesting candidates for the numerical solution of (elliptic) partial differential equations (PDEs) due to their improved approx- imation 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. 1 Introduction In this paper, we analyse quadratic FEM if applied to elliptic 2nd order PDEs. We modify the classical multigrid proof of Hackbusch/Braess to obtain a sharper result for this situation. Moreover, the analysis indicates that these results may be valid for even higher order FEM, too, leading to our conjecture that multigrid convergence rates might further improve for higher order elements. The paper is organized as follows: In section 2 we introduce our notations. We formulate the smoothing property for the multigrid algorithm, as it was already formulated by other authors [1, 3, 7], and we repeat some of the the key ingredients of the classical multigrid W-cycle proof. Section 3 specialises this proof for the situation of quadratic finite elements and points out a possible generalisation for higher order finite elements. Finally, in section 4 we perform numerical tests which show that the results of the proof are sharp. 2 Notation and key ingredients We consider a typical selfadjoint elliptic boundary value problem in a bounded domain Ω ⊂ R 2 with boundary ∂ Ω, for instance: −Δu = f in Ω,u =0 on ∂ Ω (2.1) * Institute of Applied Mathematics, University of Dortmund, Vogelpothsweg 87, D-44227 Dortmund, Ger- many, michael.koester@mathematik.uni-dortmund.de † Institute of Applied Mathematics, University of Dortmund, Vogelpothsweg 87, D-44227 Dortmund, Ger- many, stefan.turek@mathematik.uni-dortmund.de 1