Residual-free bubble finite element methods are parameter-free,
stable numerical techniques that have been successfully used
for the approximate solution of a wide range of boundary-value
problems exhibiting multiple-scale behaviour.
In the talk we present some new results concerning the RFB
method applied to convection diffusion bvp's.
Firstly, we shall discuss the RFB method on shape irregular
partitions and the implications on parameter identification
of classical stabilisation methods.
If some local features of the solution are known a priori, the
approximation properties of the RFB finite element space can be
improved through enrichment on selected edges of the partition by
edge-bubbles that are supported on pairs of neighbouring elements.
Based on this idea is the enhanced residual free bubble (RFBe)
method for the numerical approximation of convection-dominated
diffusion equations. We shall explore in two space dimensions,
both analytically and numerically, the accuracy of the RFB and
RFBe approximations focusing on the practically relevant
preasymptotic regime when the partition is not resolving the
small scales present in the solution.
The last part of the talk is dedicated to the a-posteriori
analysis of RFB, focusing on linear functional evaluation.
We shall also present an hb-adaptive algorithm in which the
bubbles enriching the finite element space are switched off
locally during the mesh adaptation process.