Optimal multilevel methods for graded bisection grids

We design and analyze optimal additive and multiplicative multilevel methods for solving H1 problems on graded grids obtained by bisection. We deal with economical local smoothers: after a global smoothing in the finest mesh, local smoothing for each added node during the refinement needs to be performed only for three vertices the new vertex and its two parent vertices. We show that our methods lead to optimal complexity for any dimensions and polynomial degree. The theory hinges on a new decomposition of bisection grids in any dimension, which is of independent interest and yields a corresponding decomposition of spaces. We use the latter to bridge the gap between graded and quasi-uniform grids, for which the multilevel theory is well-established. Mathematics Subject Classification (2000) 65M55 · 65N55 · 65N22 · 65F10 L. Chen was supported in part by NSF Grant DMS-0505454, DMS-0811272, and in part by 2010–2011 UC Irvine Academic Senate Council on Research, Computing and Libraries (CORCL). R. H. Nochetto was supported in part by NSF Grant DMS-0505454 and DMS-0807811. J. Xu was supported in part by NSF DMS-0609727, DMS 0915153, NSFC-10528102 and Alexander von Humboldt Research Award for Senior US Scientists. L. Chen (B) Department of Mathematics, University of California at Irvine, Irvine, CA 92697, USA e-mail: [email protected] R. H. Nochetto Department of Mathematics, University of Maryland, College Park, MD 20742, USA e-mail: [email protected] J. Xu The School of Mathematical Science, Peking University, Beijing, China J. Xu Department of Mathematics, Pennsylvania State University, University Park, PA 16801, USA e-mail: [email protected] 123 Author's personal copy