Virtual element method for a nonlocal elliptic problem of Kirchhoff type on polygonal meshes. (15th May 2020)

• Record Type: Journal Article
• Authors: Adak, D.; Natarajan, S.
• Is Part Of: Computers & mathematics with applications. Volume 79:issue 10(2020)
• Page Start: 2856
• Page End: 2871
• Publication Date: 2020-05-15
• Subjects: Virtual element method -- General polygonal mesh -- Error estimate -- Nonlocal elliptic equation -- Kirchhoff's equation -- Equivalent system; Electronic data processing -- Periodicals; Mathematics -- Data processing -- Periodicals; 510.28541
• DOI: 10.1016/j.camwa.2019.12.018
• Languages: English
• ISSNs: 0898-1221

Abstract

Abstract: We consider the discretization of the nonlocal elliptic problem of Kirchhoff type using the virtual element method (VEM) over polygonal meshes. The nonlocal diffusion coefficient is approximated by using the L 2 projection operator, which is directly computable from the degrees of freedom. However, the presence of the nonlocal term reduces the sparsity of the Jacobian matrix, which would increase the computational burden. To avoid this and to retain the sparsity of the Jacobian, the nonlinear system is replaced by an equivalent system inspired by the work on the FEM by Gudi (2012). The numerical results show that the proposed methodology yields optimal convergence rate in the L 2 norm. Theoretical estimates are derived for H 1 norms and are verified by solving a numerical example.