Analysis of expanded mixed finite element methods for a nonlinear parabolic equation modeling flow into variably saturated porous media
Analysis of expanded mixed finite element methods for a nonlinear parabolic equation modeling flow into variably saturated porous media
复制标题
DOI：
10.1137/s0036142996311040
复制
发表时间：
2000-03-09
影响因子：
2.9
通讯作者：
Dawson, CN
中科院分区：
文献类型：
--
作者：
Woodward, CS;Dawson, CN
本文对Richards方程的扩展混合有限元方法进行了分析。Richards方程是一个非线性抛物型偏微分方程组，它模拟了水在变饱和多孔介质中的流动。我们考虑全范围的饱和到完全不饱和的介质。在最低阶Raviart-Thomas空间和所有可能的饱和范围的情况下，我们用逼近误差的形式给出了容量误差的H+1范数。该估计使用时间积分格式和Kirchhoff变换来处理完全非饱和流动情况下的退化。然后，对于饱和到部分饱和流动的情况，给出了与容量误差有关的非线性形式的最优收敛。给出了收敛速度与容量项Holder连续性的关系。最后，给出了严格部分饱和流动情况下压力和流量的最优收敛。
We present an analysis of expanded mixed finite element methods applied to Richards' equation, a nonlinear parabolic partial differential equation modeling the flow of water into a variably saturated porous medium. We consider the full range of saturated to completely unsaturated media. In the case of the lowest order Raviart-Thomas spaces and the range of all possible saturations, we bound the H+1-norm of the error in capacity in terms of approximation error. This estimate uses a time-integrated scheme and the Kirchhoff transformation to handle a degeneracy in the case of completely unsaturated flow. Optimal convergence is then shown for a nonlinear form related to the error in the capacity for the case of saturated to partially saturated flow. Convergence rates depending on the Holder continuity of the capacity term are derived. Last, optimal convergence of pressures and fluxes is stated for the case of strictly partially saturated flow.