The article studies the homogenization of a stationary Navier-Stokes fluid in porous medium with thin film under Dirichlet boundary condition.At the end of the article,＂Darcy＇s law＂is rigorously derived from this m...The article studies the homogenization of a stationary Navier-Stokes fluid in porous medium with thin film under Dirichlet boundary condition.At the end of the article,＂Darcy＇s law＂is rigorously derived from this model as the parameter ε tends to zero,which is independent of the coordinates towards the thickness.展开更多
The porous medium equation(PME)is a typical nonlinear degenerate parabolic equation.We have studied numerical methods for PME by an energetic vari-ational approach in[C.Duan et al.,J.Comput.Phys.,385(2019),pp.13–32],...The porous medium equation(PME)is a typical nonlinear degenerate parabolic equation.We have studied numerical methods for PME by an energetic vari-ational approach in[C.Duan et al.,J.Comput.Phys.,385(2019),pp.13–32],where the trajectory equation can be obtained and two numerical schemes have been devel-oped based on different dissipative energy laws.It is also proved that the nonlinear scheme,based on f logf as the total energy form of the dissipative law,is uniquely solv-able on an admissible convex set and preserves the corresponding discrete dissipation law.Moreover,under certain smoothness assumption,we have also obtained the sec-ond order convergence in space and the first order convergence in time for the scheme.In this paper,we provide a rigorous proof of the error estimate by a careful higher or-der asymptotic expansion and two step error estimates.The latter technique contains a rough estimate to control the highly nonlinear term in a discrete W 1,∞norm and a refined estimate is applied to derive the optimal error order.展开更多
基金National NaturalScience Foundation of China(10171113 and 10471156)Natural Science Foundation of Guangdong(4009793)
文摘The article studies the homogenization of a stationary Navier-Stokes fluid in porous medium with thin film under Dirichlet boundary condition.At the end of the article,＂Darcy＇s law＂is rigorously derived from this model as the parameter ε tends to zero,which is independent of the coordinates towards the thickness.
基金The work of Yue is supported in part by NSF of China under the grants No.11971342.
文摘The porous medium equation(PME)is a typical nonlinear degenerate parabolic equation.We have studied numerical methods for PME by an energetic vari-ational approach in[C.Duan et al.,J.Comput.Phys.,385(2019),pp.13–32],where the trajectory equation can be obtained and two numerical schemes have been devel-oped based on different dissipative energy laws.It is also proved that the nonlinear scheme,based on f logf as the total energy form of the dissipative law,is uniquely solv-able on an admissible convex set and preserves the corresponding discrete dissipation law.Moreover,under certain smoothness assumption,we have also obtained the sec-ond order convergence in space and the first order convergence in time for the scheme.In this paper,we provide a rigorous proof of the error estimate by a careful higher or-der asymptotic expansion and two step error estimates.The latter technique contains a rough estimate to control the highly nonlinear term in a discrete W 1,∞norm and a refined estimate is applied to derive the optimal error order.
