Some multilevel methods on graded meshes

  • Authors:

  • M. Jung;S. Nicaise;J. Tabka

  • Affiliations:

  • Fakultät für Mathematik und Naturwissenschaften, Institut für Wissenschaftliches Rechnen, Technische Universität Dresden, D-01062 Dresden, Germany;Université de Valenciennes et du Hainaut Cambrésis, MACS, F-59313 Valenciennes Cedex 9, France;Université de Valenciennes et du Hainaut Cambrésis, MACS, F-59313 Valenciennes Cedex 9, France

  • Venue:
  • Journal of Computational and Applied Mathematics
  • Year:
  • 2002

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Abstract

We consider Yserentant's hierarchical basis method and multilevel diagonal scaling method on a class of refined meshes used in the numerical approximation of boundary value problems on polygonal domains in the presence of singularities. We show, as in the uniform case, that the stiffness matrix of the first method has a condition number bounded by (ln(1/h))2, where h is the meshsize of the triangulation. For the second method, we show that the condition number of the iteration operator is bounded by ln(1/h), which is worse than in the uniform case but better than the hierarchical basis method. As usual, we deduce that the condition number of the BPX iteration operator is bounded by ln(1/h). Finally, graded meshes fulfilling the general conditions are presented and numerical tests are given which confirm the theoretical bounds.