SUPG stabilization for the nonconforming virtual element method for advection-diffusion-reaction equations

TL;DR

This paper introduces a stabilized nonconforming virtual element method with SUPG for convection-diffusion-reaction equations, providing convergence analysis and numerical validation in convection-dominated regimes.

Contribution

It develops a novel VEM-SUPG approach with exact polynomial projections and stability considerations, enhancing the method's accuracy and robustness for convection-dominated problems.

Findings

Optimal convergence rates are achieved and confirmed numerically.

The method effectively stabilizes solutions in convective regimes.

Theoretical analysis supports the numerical results.

Abstract

We present the design, convergence analysis and numerical investigations of the nonconforming virtual element method with Streamline Upwind/Petrov-Galerkin (VEM-SUPG) stabilization for the numerical resolution of convection-diffusion-reaction problems in the convective-dominated regime. According to the virtual discretization approach, the bilinear form is split as the sum of a consistency and a stability term. The consistency term is given by substituting the functions of the virtual space and their gradients with their polynomial projection in each term of the bilinear form (including the SUPG stabilization term). Polynomial projections can be computed exactly from the degrees of freedom. The stability term is also built from the degrees of freedom by ensuring the correct scalability properties with respect to the mesh size and the equation coefficients. The nonconforming formulation relaxes the continuity conditions at cell interfaces and a weaker regularity condition is considered involving polynomial moments of the solution jumps at cell interface. Optimal convergence properties of the method are proved in a suitable norm, which includes a contribution from the advective stabilization terms. Experimental results confirm the theoretical convergence rates.