Workshop on Numerical Methods and Analysis in CFD - Abstract
Kopteva, Natalia
Standard and stabilized finite element approximations are considered on shape-regular meshes for singularly perturbed convection-diffusion equations. Our initial result is that natural maximum-norm residual estimators reliably control the error in the maximum norm, assuming suitable estimates of the Green's function hold. On the other hand, residual-type estimators in the energy norm are only efficient up to a dual norm of the convective error (as shown by R. Verfürth). A main contribution of the paper is to analogously define a suitable dual seminorm of the convective error. Having defined such a dual norm, we then define the total error as the originally targeted maximum norm of the error plus the dual seminorm of the convective error plus standard data oscillation terms. Our a posteriori error estimator is then shown to be equivalent to the total error (up to a logarithmic factor). Numerical experiments illustrate the behavior and performance of our estimators in the context of uniform and adaptive mesh refinement. In particular, they show that the estimators may vastly overestimate the error in the maximum norm alone, but they closely track the total error as predicted by our theory. Adaptive refinement based on our error indicators is also shown to do an effective job at automatically resolving standard model problems whose solutions include strong layers.
This is joint work with Alan Demlow and Sebastian Franz.