Virtual element methods for nonlocal parabolic problems on general type of meshes. (English) Zbl 1471.65131
Summary: 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 \(L^2\) 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.
MSC:
| 65M60 | Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs |
| 65M06 | Finite difference methods for initial value and initial-boundary value problems involving PDEs |
| 65N30 | Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs |
| 65H10 | Numerical computation of solutions to systems of equations |
| 65N12 | Stability and convergence of numerical methods for boundary value problems involving PDEs |
| 65N15 | Error bounds for boundary value problems involving PDEs |
| 35A01 | Existence problems for PDEs: global existence, local existence, non-existence |
Keywords:
virtual element method; polygonal/polyhedral meshes; error estimates; nonlocal parabolic equation; nonlinear equationsReferences:
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