Localization of the W^{-1,q} and distance norms

 

p_1_5_liftings

Distribution of the global (left) and local (right) p-Laplace residual lifting, p = 1.5

p_10_liftings

Distribution of the global (left) and local (right) p-Laplace residual lifting, p = 10

eff_ind_-001

Effectivity index, transmission problem, left subdomain diffusion -0.01

eff_ind_-033

Effectivity index, transmission problem, left left subdomain diffusion -0.33

eff_ind_-099

Effectivity index, transmission problem, left left subdomain diffusion -0.99

Main results:

  • dual norm of any bounded linear functional on the Sobolev space W01,p localizes under an orthogonality condition: equals the lq sum of local contributions;
  • estimates taking into account the violation of the orthogonality condition;
  • distance to the Sobolev space H01 localizes: equals the l2 sum of local contributions (no orthogonality condition);
  • implies local efficiency and robustness of a posteriori estimates for nonlinear and non-coercive partial differential equations in divergence form;
  • includes the case of inexact solvers.

Details:

H01 setting including nonconformity: paper (preprint) with Patrick Ciarlet, Jr., presentation

W01,p setting: paper (preprint) with Jan Blechta and Josef Málek, presentation

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