Fluctuation-based stabilization (I)

Abstract

In this chapter, we still use \(H^1\)-conforming finite elements and a boundary penalty technique, but we consider a different stabilization technique. One motivation is that the residual-based stabilization from the previous chapter is delicate to use when approximating time-dependent PDEs since the time derivative is part of the residual. The techniques devised in this chapter and the next one avoid this difficulty. The starting observation is that \(H^1\)-conforming test functions cannot control the gradient of \(H^1\)-conforming functions since the gradient generally exhibits jumps across the mesh interfaces. The idea behind fluctuation-based stabilization is to gain full control on the gradient by adding a least-squares penalty on the part of the gradient departing from the \(H^1\)-conforming space, and this part can be viewed as a fluctuation. Stabilization techniques based on this idea include the continuous interior penalty (CIP) method, studied in this chapter, and two-scale stabilization techniques such as the local projection stabilization (LPS) and the subgrid viscosity (SGV) methods, which are studied in the next chapter. We present in this chapter a unified analysis based on an abstract set of assumptions. We show in this chapter and the next one how to satisfy these assumptions using CIP, LPS, and SGV.