Proper intrinsic scales for a-posteriori multiscale error estimation

Recently the multiscale a-posteriori error estimator has been introduced, showing excellent robustness for fluid mechanics problems. In this paper, a theoretical analysis for element edge exact solutions is conducted, in which case, the error constant is the norm of a Green’s function or a residual-free bubble. This finds application when the solution is computed with a stabilized method. One of the features of the technique is that it gives the proper scales for a-posteriori error estimation in any norm of interest, such as the L2, H1, energy and L∞ norms. For fluid transport problems it is shown that the constant for predicting the error in the H1 seminorm is unbounded as the element Peclet number tends to infinity, making Lp norms more suitable for this type of problems. Furthermore, it is shown that the flow intrinsic time scale parameter represents the L1 norm of the error as a function of the L∞ norm of the residual. When these scales are employed for one-dimensional nodally-exact solutions, piecewise linear finite element spaces and piecewise constant residuals, the multiscale error estimator is shown to be exact.