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Numerical Methods for PDEs: In Occasion of Raytcho Lazarov's 70th Birthday

Svetozar Margenov, Bulgarian Academy of Sciences
Robust Multilevel Methods for Strongly Heterogeneous Problems


The first part of this talk is devoted to the construction and analysis of hierarchical basis algebraic multilevel iteration (AMLI) methods in the case of coefficient jumps which are aligned with the interfaces of the initial mesh. The condition number estimates are uniform with respect to both mesh and/or coefficient anisotropy, the coefficient jumps, as well as the size of the discrete problem. The computational complexity is proportional to the number of degrees of freedom.

Robust multilevel methods for high-frequency and high-contrast problems are presented in the second part. Some advantages of the nonlinear AMLI methods including the case of element-by-element approximation of the Schur compliment are discussed.

The numerical tests demonstrate:

  1. robustness of the convergence estimates
  2. recent scalability results beyond some assumptions of the rigorous theory