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; L2)-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 Lp-regularity of fully discrete finite element solutions of parabolic equations, optimal error estimate in Lp (0 , T; Lq)-norm and almost optimal error estimate in L∞ (0 , T; Lq)-norm are established under the assumption of D(u) being Lipschitz continuous with respect to u.
| Original language | English |
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| Pages (from-to) | 1009-1042 |
| Journal | Numerische Mathematik |
| Volume | 141 |
| Issue number | 4 |
| Online published | 28 Feb 2019 |
| DOIs | |
| Publication status | Published - Apr 2019 |
Funding
Wentao Cai: The research of this author was supported in part by a Hong Kong RGC Grant (15301818). Buyang Li: The research of this author was supported in part by an internal grant of The Hong Kong Polytechnic University (project code 1-ZE6L) and a Hong Kong RGC Grant (15301818). Yanping Lin: The research of this author was supported in part by a Hong Kong RGC Grant (15302418). Weiwei Sun: The research of this author was supported in part by the Zhujiang Scholar program, a grant from South China Normal University and a Hong Kong RGC Grant (CityU 11300517).