SUPG stabilization for the nonconforming virtual element method for advection-diffusion-reaction equations
This work provides a novel numerical method for solving convection-dominated problems, which are challenging for standard finite element methods, but the contribution is incremental as it extends existing VEM-SUPG ideas to the nonconforming setting.
The paper introduces a nonconforming virtual element method with SUPG stabilization for advection-diffusion-reaction equations in the convection-dominated regime, proving optimal convergence rates in a suitable norm and confirming them numerically.
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.