Numerical solution of nonlinear convection-diffusion-reaction equation using a stabilized virtual element method

Abstract

The virtual element method (VEM) was proposed for the nonlinear convection-diffusion-reaction problem in [5]. Using projection operators, a computable VEM discrete scheme was derived and the existence of the solution was proved. However, even when higher order elements were introduced, the SUPG framework shows spurious oscillations in the crosswind direction. In this paper, we propose, in the context of VEM, a shock capture technique inspired by the work of [30] to provide a stable and more robust solution technique that does not exhibit numerical oscillations when higher order elements are employed. For the proposed framework, convergence analysis is performed and optimal order error estimates are derived in the energy norm. Numerical experiments are computed to show the performance of this technique and to validate the theoretical results obtained.