Nonoverlapping Additive Schwarz Method for hp-DGFEM with Higher-order Penalty Terms

摘要

Let us consider a second order elliptic equation -div((∂)▽u)=f in Ω and u=0 in ∂Ω.(1) The problem is discretized by an h-p symmetric interior higher-order [4] discontinuous Galerkin finite element method. In a K-th order multipenalty method, one penalizes the jumps of scaled normal higher-order derivatives up to order K across the interelement boundaries - so the standard interior penalty method corresponds to taking K = 0. The idea to penalize the discontinuity in the flux (K = 1) of the discrete solution was introduced by Douglas and Dupont [6]. It addresses the observation that the flux (which is an important quantity in many applications) of the accurate solution is continuous. Giving the user a possibility to control the inevitable violation of this principle makes the discretization method more robust and conservative. Recently, flux jump penalization has been used to improve stability properties of an unfitted Nitsche's method [5], the case K > 1 was also considered in [1] for the immersed finite element method to obtain higher-order discretizations.