Convergence Analysis of a Numerical Scheme for the Porous Medium Equation by an Energetic Variational Approach  

作　　者：Chenghua Duan Chun Liu Cheng Wang Xingye Yue 

机构地区：[1]Department of Mathematics,Soochow University,Suzhou 215006,China [2]Shanghai Center for Mathematical Sciences,Fudan University,Shanghai 200438,China [3]Department of Applied Mathematics,Illinois Institute of Technology,Chicago,IL 60616,USA [4]Department of Mathematics,University of Massachusetts,Dartmouth,North Dartmouth,MA 02747-2300,USA

出　　处：《Numerical Mathematics(Theory,Methods and Applications)》2020年第1期63-80,共18页高等学校计算数学学报（英文版）

基　　金：The work of Yue is supported in part by NSF of China under the grants No.11971342.

摘　　要：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.

关 键 词：Energetic variational approach porous medium equation trajectory equation optimal rate convergence analysis 

分 类 号：O17[理学—数学]