Lowest order stabilization free Virtual Element Method for the 2D Poisson equation
This work addresses the stabilization issue in virtual element methods for Poisson problems, offering a simpler formulation without sacrificing accuracy.
The paper introduces the first order Enlarged Enhancement Virtual Element Method (E$^2$VEM) for the 2D Poisson equation, which eliminates the need for a stabilization term by using higher order polynomial projections. Numerical tests confirm optimal convergence rates on convex and non-convex polygonal meshes.
We introduce and analyse the first order Enlarged Enhancement Virtual Element Method (E$^2$VEM) for the Poisson problem. The method allows the definition of bilinear forms that do not require a stabilization term, thanks to the exploitation of higher order polynomial projections that are made computable by suitably enlarging the enhancement (from which comes the prefix of the name E$^2$) property of local virtual spaces. The polynomial degree of local projections is chosen based on the number of vertices of each polygon. We provide a proof of well-posedness and optimal order a priori error estimates. Numerical tests on convex and non-convex polygonal meshes confirm the criterium for well-posedness and the theoretical convergence rates.