A polygonal discontinuous Galerkin method with minus one stabilization

Bertoluzza, Silvia; Prada, Daniele

ESAIM: Mathematical Modelling and Numerical Analysis , Tome 55 (2021), pp. S785-S810

We propose a discontinuous Galerkin method for the Poisson equation on polygonal tessellations in two dimensions, stabilized by penalizing, locally in each element K, a residual term involving the fluxes, measured in the norm of the dual of H¹ (K). The scalar product corresponding to such a norm is numerically realized via the introduction of a (minimal) auxiliary space inspired by the Virtual Element Method. Stability and optimal error estimates in the broken H¹ norm are proven under a weak shape regularity assumption allowing the presence of very small edges. The results of numerical tests confirm the theoretical estimates.

Classification : 65N12, 65N30
Keywords: Discontinuous Galerkin method, polygonal tessellation, minus one stabilization

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