A Polygonal Discontinuous Galerkin Method with Minus One Stabilization

We propose an Hybridized Discontinuous Galerkin method on polygonal tessellations, stabilized by penalizing, locally in each element $K$, a residual term involving the fluxes, measured in the norm of the dual of $H^1(K)$. The scalar product corresponding to such a norm is numerically realized via the introduction of a (minimal) auxiliary space of VEM type. Stability and optimal error estimates in the broken $H^1$ 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.Comment: 27 pages, 2 figure

An abstract framework for heterogeneous coupling: stability, approximation and applications

Introducing a coupling framework reminiscent of FETI methods, but here on abstract form, we establish conditions for stability and minimal requirements for well-posedness on the continuous level, as well as conditions on local solvers for the approximation of subproblems. We then discuss stability of the resulting Lagrange multiplier methods and show stability under a mesh conditions between the local discretizations and the mortar space. If this condition is not satisfied we show how a stabilization, acting only on the multiplier can be used to achieve stability. The design of preconditioners of the Schur complement system is discussed in the unstabilized case. Finally we discuss some applications that enter the framework