两项时间混合分数阶扩散波动方程的有限元高精度分析  

High-accuracy analysis of finite-element method for two-term mixed time-fractional diffusion-wave equations

作　　者：魏亚冰 赵艳敏[1] 唐贻发[3,4] 王芬玲 史争光[5] 李匡郢 Yabing WEI1,2, Yanmin ZHAO1, Yifa TANG3,4, Fenling WANG1, Zhengguang SHI5 ＆ Kuangyin LI6(1. School of Mathematics and Statistics, Xuchang University, Xuchang 461000, China; 2. School of Mathematical and Statistics, Zhengzhou University, Zhengzhou 450001, China; 3. LSEC, ICMSEC, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190 China; 4. School of Mathematical Sciences, University of Chinese Academy of Sciences, Beijing 100049, China; 5. School of Economic Mathematics, Southwestern University of finance and Economic, Chengdu 611130, China; 6. McDougall School of Petroleum Engineering, The University of Tulsa, Tulsa, OK 74104, US)

机构地区：[1]许昌学院数学与统计学院 [2]郑州大学数学与统计学院 [3]中国科学院数学与系统科学研究院 [4]中国科学院大学数学科学学院 [5]西南财经大学经济数学学院 [6]McDougall School of Petroleum Engineering, The University of Tulsa

出　　处：《中国科学：信息科学》2018年第7期871-887,共17页Scientia Sinica(Informationis)

基　　金：国家自然科学基金(批准号:11771438;11101381;11471296)资助项目

摘　　要：基于空间方向的有限元方法和时间方向的L1-CN格式,本文针对二维两项时间混合分数阶扩散波动方程进行数值分析.首先,给出该方程的全离散逼近格式,并证明其无条件稳定性.然后,严格证明L^2模意义下的收敛结果和H^1模意义下的超逼近结果O(h^2+τ^(min{2-α)1,^(3-α}))(0<α_1<1,1<α<2),这里h和τ分别表示空间和时间步长.进一步地,利用插值后处理技术导出H^1模意义下的整体超收敛结果.最后,借助于数值算例进一步展示理论分析的正确性和高效性.Based on the finite-element method（FEM） in the spatial direction and L1-CN approximation in the temporal direction, respectively, numerical analyses are proposed for 2D two-term mixed time-fractional diffusionwave equations. First, a fully discrete approximate scheme is established for the equation, and it is proved to be unconditionally stable. Then, rigorous proofs are provided for the convergence result in the L^2-norm and superclose properties in H^1-norm with order O（h^2＋ τ^（min{2-α1,3-α}（）（0 α1 1, 1 α 2）, where h and τ are the spatial size and time step, respectively. Furthermore, the global superconvergence in the H^1-norm is obtained using an interpolation postprocessing technique. Finally, with the help of numerical examples, the correctness and high efficiency of the theoretical analysis are further demonstrated.

关 键 词：分数阶扩散波动方程 有限元方法 L1-CN格式 稳定性 超逼近 收敛和超收敛 

分 类 号：O241.82[理学—计算数学]