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Virtual element methods for nonlocal parabolic problems on general type of meshes

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Abstract

In this paper, we consider the discretization of a parabolic nonlocal problem within the framework of the virtual element method. Using the fixed point argument, we prove that the fully discrete scheme has a unique solution. The presence of the nonlocal term makes the problem nonlinear, and the resulting nonlinear equations are solved using the Newton method. The computational cost of the Jacobian of the nonlinear scheme increases in the presence of nonlocal coefficient. To reduce the computational burden in computing the Jacobian, which otherwise is inevitable in the usual approach, in this paper, we propose an equivalent formulation. A priori error estimates in the L2 and the H1 norms are derived. Furthermore, we employ a linearized scheme without compromising the rate of convergence in the respective norms. Finally, the theoretical convergence results are verified through numerical experiments over polygonal meshes.

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Acknowledgments

The authors are deeply grateful to Prof. David Mora (Universidad del Bío-Bío, Concepción, Chile) for the fruitful discussions. We convey our sincere gratitude to Arrutselvi M and E. Natarajan (Indian Institute of Space Science and Technology-Thiruvananthapuram) for helping the implementation of the VEM in Matlab.

Funding

Dibyendu Adak was partially supported by CONICYT-Chile through FONDECYT Postdoctorado project 3200242, Departamento de Matemática, Universidad del Bío-Bío, Chile.

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Authors and Affiliations

  1. Department of Mechanical Engineering, Indian Institute of Technology Madras, Chennai, 600036, India

    D. Adak & S. Natarajan

  2. GIMNAP, Departamento de Matemática, Universidad del Bío-Bío, Concepción, Chile

    D. Adak

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  1. D. Adak
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  2. S. Natarajan
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Correspondence to S. Natarajan.

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Communicated by: Lourenco Beirao da

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Adak, D., Natarajan, S. Virtual element methods for nonlocal parabolic problems on general type of meshes. Adv Comput Math 46, 74 (2020). https://doi.org/10.1007/s10444-020-09811-0

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