Virtual element methods for nonlocal parabolic problems on general type of meshes
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Virtual element methods for nonlocal parabolic problems on general type of meshes D. Adak1,2 · S. Natarajan1 Received: 14 February 2020 / Accepted: 3 August 2020 / © Springer Science+Business Media, LLC, part of Springer Nature 2020
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 H 1 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. Keywords Virtual element method · Polygonal/polyhedral meshes · Error estimates · Nonlocal parabolic equation · Nonlinear equations Mathematics Subject Classification (2010) 65N30 · 65N12 · 65M60 · 65N15
1 Introduction We present a virtual element formulation for the following nonlocal parabolic problem: ⎧ (x, t) ∈ × (0, T ), ⎪ ⎨ ut − H(u) u =f (u) u(x, 0) =u0 (x) x ∈ , (1.1) ⎪ ⎩ u(x, t) =0 (x, t) ∈ ∂ × (0, T ), Communicated by: Lourenco Beirao da S. Natarajan
[email protected] 1
Department of Mechanical Engineering, Indian Institute of Technology Madras, Chennai, 600036, India
2
GIMNAP, Departamento de Matem´atica, Universidad del B´ıo-B´ıo, Concepci´on, Chile
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Adv Comput Math
where H(u) := H
(2020) 46:74
u d is the nonlocal diffusive coefficient, a function of
u d [1], T > 0 is a final time and ⊂ Rd (d = 2, 3) is a bounded polygonal/polyhedral domain with Lipschitz boundary ∂. Equation 1.1 is not pointwise identity, and hence this problem is called nonlocal. The nonlocal parabolic problem has encountered considerable attention due to its wide application in modelling many physical and biological phenomena [2–5]. Existing approaches to solve the model problem (1.1) typically employ Galerkin-based finite element method (FEM). The FEM relies on discretizing the domain with non-overlapping regions and employ polynomials to represent the unknown field variables. We observe that in this case the domain discretization is restricted to simplex elements, such as triangles/quadrilateral elements in two dimensions and tetrahedra/hexahedra in three dimensions. Recently, the focus has been to develop methods that can make use of general polytopal meshes. This is because these methods reduce the meshing burden and easily accommodate elements with hanging nodes in a quadtree/octree decomposition. Some of the methods th
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