Patch Bubbles for Advection‐Dominated Steady and Unsteady Problems

ABSTRACT A novel variant of the 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 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 enhanced residual‐free bubble (eRFB) method by Cangiani and Süli [7, 8], 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.