Stabilizing DG methods on polygonal meshes via computable dual norms

Istituto Eulero

Data: 25 Maggio 2022 / 16:30 - 17:30

USI Campus EST, room C1.04, Sector C // Online on Zoom

Please click here to join (Meeting-ID: 839 7179 3831 / Kenncode: 730237).

Speaker: Silvia Bertoluzza, Italian National Research Council, Pavia.

In many problems from different applications, a partial differential equation arises for which the linear differential operatorn is not coercive but only positive semidefinite. For instance, this is the case of the Stokes equation, and of the equations arising from the application the Discontinuous Galerkin method to an elliptic PDE. It is well known that when discretizing equations of such a form, instability problems can arise. The remedy is either to choose the discretization space carefully, which might be difficult or even practically unfeasible, or to utilize some stabilization technique, allowing one to transform the unstable discrete problem into a stable one by adding either suitable elements to the discretization space, or (and this is the case that we are going to consider), suitable consistent terms to the equation itself, which penalize some residual, usually in a mesh dependent norm. Estimates on the resulting methods rely on the existence of direct and/or inverse inequalities relating such norm to the norm of the dual space where the residual naturally lives. In this talk we discuss the possibility of designing computable dual norms, and of using them in the design of stabilization terms. To illustrate this idea, we will propose a DG method for solving the Poisson equation on polygonal meshes, in which the unknown in the polygonal elements, the fluxes and the unknown on the edges, are all, independently, approximated by polynomials of degree k. Well posedness is achieved using a suitable minus one norm stabilization term penalizing the discrepancy between actual fluxes of the solution in the element and the unknown that independently discretizes such fluxes.

Silvia Bertoluzza is Dirigente di ricerca at the Institute for Numerical Analysis and Information Technologies of the italian National Research Council, in Pavia. Her research activity is mainly focused on the theoretical analysis of numerical methods for the solution of partial differential equations. In this framework she studies several aspect of different approaches, such as domain decomposition and fictitious domain methods,  stabilization techniques, wavelet methods, polytopal methods.  She has been Network coordinator for the EC TMR Network "Wavelets in numerical simulation" and of the EC IHP Network "Breaking Complexity", and she is currently unit responsible for the ERC Advanced Grant  "New CHallenges for (adaptive) PDE solvers: the interplay of ANalysis and GEometry" (PI: Annalisa Buffa), active from October 2016 to September 2022.