A Posteriori Modelling-Discretization Error Estimate for Elliptic Problems with L ∞-Coefficients
Weymuth, M., Sauter, S., & Repin, S. (2017). A Posteriori Modelling-Discretization Error Estimate for Elliptic Problems with L ∞-Coefficients. Computational Methods in Applied Mathematics, 17(3). https://doi.org/10.1515/cmam-2017-0015
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Computational Methods in Applied MathematicsDate
2017Copyright
© 2017 Walter de Gruyter GmbH, Berlin/Boston. This is a final draft version of an article whose final and definitive form has been published by de Gryuter. Published in this repository with the kind permission of the publisher.
We consider elliptic problems with complicated, discontinuous diffusion tensor A0.
One of the standard approaches to numerically treat such problems is to simplify the
coefficient by some approximation, say Aε, and to use standard finite elements. In [19]
a combined modelling-discretization strategy has been proposed which estimates the
discretization and modelling errors by a posteriori estimates of functional type. This
strategy allows to balance these two errors in a problem adapted way. However, the
estimate of the modelling error is derived under the assumption that the difference
A0 − Aε is bounded in the L∞-norm, which requires that the approximation of the
coefficient matches the discontinuities of the original coefficient. Therefore this theory is
not appropriate for applications with discontinuous coefficients along complicated, curved
interfaces. Based on bounds for A0 − Aε in an L
q
-norm with q < ∞ we generalize the
combined modelling-discretization strategy to a larger class of coefficients.
...
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