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Title: Analysis of fully discrete FEM for miscible displacement in porous media with Bear–Scheidegger diffusion tensor
Authors: Cai, W
Li, B 
Lin, Y 
Sun, W
Issue Date: Apr-2019
Source: Numerische mathematik, Apr. 2019, v. 141, no. 4, p. 1009-1042
Abstract: Fully discrete Galerkin finite element methods are studied for the equations of miscible displacement in porous media with the commonly-used Bear–Scheidegger diffusion–dispersion tensor: D(u)=γdmI+|u|(αTI+(αL-αT)u⊗u|u|2).Previous works on optimal-order L ∞ (0 , T; L 2 ) -norm error estimate required the regularity assumption ∇ x ∂ t D(u(x, t)) ∈ L ∞ (0 , T; L ∞ (Ω)) , while the Bear–Scheidegger diffusion–dispersion tensor is only Lipschitz continuous even for a smooth velocity field u. In terms of the maximal L p -regularity of fully discrete finite element solutions of parabolic equations, optimal error estimate in L p (0 , T; L q ) -norm and almost optimal error estimate in L ∞ (0 , T; L q ) -norm are established under the assumption of D(u) being Lipschitz continuous with respect to u.
Publisher: Springer
Journal: Numerische mathematik 
ISSN: 0029-599X
EISSN: 0945-3245
DOI: 10.1007/s00211-019-01030-0
Rights: © Springer-Verlag GmbH Germany, part of Springer Nature 2019
This is a post-peer-review, pre-copyedit version of an article published in Numerische Mathematik. The final authenticated version is available online at: http://dx.doi.org/10.1007/s00211-019-01030-0
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