Fully discrete finite element scheme for nonlocal parabolic problem involving the Dirichlet energy

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• 作者：Vimal Srivastava ; Sudhakar Chaudhary…
• 关键词：Nonlocal ; Kirchhoff equation ; Backward Euler method ; Newton iteration method
• 刊名：Journal of Applied Mathematics and Computing
• 出版年：2017
• 出版时间：February 2017
• 年：2017
• 卷：53
• 期：1-2
• 页码：413-443
• 全文大小：
• 刊物类别：Mathematics and Statistics
• 刊物主题：Computational Mathematics and Numerical Analysis; Appl.Mathematics/Computational Methods of Engineering; Theory of Computation; Mathematics of Computing;
• 出版者：Springer Berlin Heidelberg
• ISSN：1865-2085
• 卷排序：53

文摘

In this article we present a finite element scheme for solving a nonlocal parabolic problem involving the Dirichlet energy. For time discretization, we use backward Euler method. The nonlocal term causes difficulty while using Newton’s method. Indeed, after applying Newton’s method we get a full Jacobian matrix due to the nonlocal term. In order to avoid this difficulty we use the technique given by Gudi (SIAM J Numer Anal 50(2):657–668, 2012) for elliptic nonlocal problem of Kirchhoff type. We discuss the well-posedness of the weak formulation at continuous as well as at discrete levels. We also derive a priori error estimates for both semi-discrete and fully discrete formulations. Results based on the usual finite element method are provided to confirm the theoretical estimates.