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Virtual element method for semilinear sine–Gordon equation over polygonal mesh using product approximation technique

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  • Adak, D.
  • Natarajan, S.

Abstract

In this paper, we employ the linear virtual element spaces to discretize the semilinear sine–Gordon equation in two dimensions. The salient features of the virtual element method (VEM) are: (a) it does not require explicit form of the shape functions to construct the nonlinear and the bilinear terms, and (b) relaxes the constraint on the mesh topology by allowing the domain to be discretized with general polygons consisting of both convex and concave elements, and (c) easy mesh refinements (hanging nodes and interfaces are allowed). The nonlinear source term is discretized by employing the product approximation technique and for temporal discretization, the Crank–Nicolson scheme is used. The resulting nonlinear equations are solved using the Newton’s method. We derive a priori error estimations in L2 and H1 norms. The convergence properties and the accuracy of the virtual element method for the solution of the sine–Gordon equation are demonstrated with academic numerical experiments.

Suggested Citation

  • Adak, D. & Natarajan, S., 2020. "Virtual element method for semilinear sine–Gordon equation over polygonal mesh using product approximation technique," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 172(C), pages 224-243.
    • Handle: RePEc:eee:matcom:v:172:y:2020:i:c:p:224-243
    DOI: 10.1016/j.matcom.2019.12.007
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    References listed on IDEAS

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    1. Dehghan, Mehdi & Shokri, Ali, 2008. "A numerical method for solution of the two-dimensional sine-Gordon equation using the radial basis functions," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 79(3), pages 700-715.
    2. Sheng, Q. & Khaliq, A.Q. M. & Voss, D.A., 2005. "Numerical simulation of two-dimensional sine-Gordon solitons via a split cosine scheme," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 68(4), pages 355-373.
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