Patch Bubbles for Advection‐Dominated Steady and Unsteady Problems

Abstract

A novel variant of the emph{residual-free bubble} method (RFB) for advection dominated problems is presented. Since the usual RFB still suffers from oscillations and strong under/overshoots, the bubble space is enriched by emph{patch bubbles}, giving more freedom to the bubble space.We use a recursive and efficient approach to accurately compute the bubbles. Numerical experiments clearly demonstrate the superiority of our method compared to the standard RFB.While similar in spirit to the emph{enhanced residual-free bubble} (eRFB) method by Cangiani and S"uli~cite{Cangiani2005long,Cangiani2005short}, our approach differs in the definition and computation of the additional bubbles.In addition, we extend the methodology to solve problems with non constant coefficients and we develop a novel bubble-based stabilization technique for time-dependent problems, which performs very accurately.