Analysis of fully discrete FEM for miscible displacement in porous media with Bear–Scheidegger diffusion tensor
/ Authors
/ 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: $$\begin{aligned} D(\mathbf{u}) = \gamma d_m I + |\mathbf{u}|\bigg ( \alpha _T I + (\alpha _L - \alpha _T) \frac{\mathbf{u} \otimes \mathbf{u}}{|\mathbf{u}|^2}\bigg ) \, . \end{aligned}$$D(u)=γdmI+|u|(αTI+(αL-αT)u⊗u|u|2).Previous works on optimal-order $$L^\infty (0,T;L^2)$$L∞(0,T;L2)-norm error estimate required the regularity assumption $$\nabla _x\partial _tD(\mathbf{u}(x,t)) \in L^\infty (0,T;L^\infty (\Omega ))$$∇x∂tD(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 $$\mathbf{u}$$u. In terms of the maximal $$L^p$$Lp-regularity of fully discrete finite element solutions of parabolic equations, optimal error estimate in $$L^p(0,T;L^q)$$Lp(0,T;Lq)-norm and almost optimal error estimate in $$L^\infty (0,T;L^q)$$L∞(0,T;Lq)-norm are established under the assumption of $$D(\mathbf{u})$$D(u) being Lipschitz continuous with respect to $$\mathbf{u}$$u.
Journal: Numerische Mathematik