Virtual element method for the quasilinear convection-diffusion-reaction equation on polygonal meshes. (English) Zbl 1501.65102
Summary: In this paper, we analyze the virtual element method for the quasilinear convection-diffusion-reaction equation. The most important part in the analysis is the proof of existence and uniqueness of the branch of solution of the discrete problem. We extend the explicit analysis given by G. Lube [Numer. Math. 61, No. 3, 335–357 (1992; Zbl 0759.65076)] for the finite element discretization to virtual element framework. We prove the optimal rate of convergence in the energy norm. In order to reduce the overall computational cost incurred for the nonlinear equations, we have performed the numerical experiments using a two-grid method. We validate the theoretical estimates with the computed numerical results.
MSC:
| 65N30 | Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs |
| 65N55 | Multigrid methods; domain decomposition for boundary value problems involving PDEs |
| 65N12 | Stability and convergence of numerical methods for boundary value problems involving PDEs |
| 65N15 | Error bounds for boundary value problems involving PDEs |
Citations:
Zbl 0759.65076Software:
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