Comput. Mech. (2006) DOI 10.1007/s00466-006-0048-7 ORIGINAL PAPER Guillermo Hauke · Mohamed H. Doweidar Daniel Fuster · Antonio Gómez · Javier Sayas Application of variational a-posteriori multiscale error estimation to higher-order elements Received: 31 October 2005 /Accepted: 09 February 2006 © Springer-Verlag 2006 Abstract An explicit a-posteriori error estimator based on the variational multiscale method is extended to higher-order elements. The technique is based on a recently derived explicit formula of the fine-scale Green’s function for higher- order elements. For the class of element-edge exact meth- ods, the technique is able to predict the error exactly in any desired norm. It is shown that for elements of order k , the exact error depends on the k - 1 derivative of the residual. The technique is applied to one-dimensional examples of fluid transport computed with stabilized methods. Keywords A posteriori error estimation · Advection-diffu- sion equation · Hyperbolic flows · Fluid mechanics · Fluid dynamics · Stabilized methods · Variational multiscale method 1 Introduction The variational multiscale method [17,18] offers a fresh the- oretical framework to study and develop a-posteriori error estimators [1]. This idea has been initially explored in [13] to identify an explicit a-posteriori error estimator for stabilized solutions, especially well suited for convection-dominated flows. Practical applications show that the adaptive meshes generated with this technique are of high quality. The above paper was followed by [14,12], where the proper intrinsic scales for a posteriori error estimation were derived for the class of element-edge exact methods, i.e. methods characterized by solutions which are exact along the element boundaries. Clearly, for methods where the error is zero along the ele- ment edges, the error propagation is stopped at the interele- G. Hauke (B ) · M. H. Doweidar · D. Fuster · A. Gómez Departamento de Mecánica de Fluidos, Centro Politécnico Superior, C/Maria de Luna 3, 50018 Zaragoza, Spain E-mail: ghauke@unizar.es J. Sayas Departamento de Matemática Aplicada, Centro Politécnico Superior, C/Maria de Luna 3, 50018 Zaragoza, Spain ment boundaries, and the error remains confined within each element. The operator that distributes the error in a numerical solution is the fine-scale Green’s function, which takes into consideration both, the error propagation and the projection of the exact solution into the finite element space. Furthermore, Hughes and Sangalli [19] show that in the one-dimensional case and for methods based on the H 1 0 pro- jector, although the Green’s function is a global function, the fine-scale Green’s function is local and confined within the elements. In the multi-dimensional setting and for advection- dominated flows (which corresponds to the hyperbolic limit) [19] also shows that the above attributes approximately hold for methods based on the H 1 0 projector. Thus, the work here relies of these findings. One of the traits of this framework is that it gives the rec- ipe to calculate the error constants in the desired error norm. In this context, it was also shown that the classical intrinsic time-scale parameter [6,9,17] carries error information in the L 1 norm as a function of the L ∞ residual norm. Since the error is propagated according to the fine-scale Green’s function, as the nature of this function is better under- stood, the above mentioned results can be extended to a wider class of problems. For instance, the work by Hughes and San- galli [19] develops explicit formulas for higher-order fine- scale Green’s functions, which are exploited in this paper to extend the previous work on variational multiscale a-poste- riori error estimation to higher-order elements. 2 The variational multiscale approach to error estimation 2.1 The abstract problem Consider a spatial domain  with boundary Ŵ. The strong form of the boundary-value problem consists of finding u :  → R such that for the given essential boundary condition g : Ŵ g → R, the natural boundary condition h : Ŵ h → R, and forcing function f :  → R, f ∈ L 2 (if Ŵ h =∅, f ∈ H -1 ), the following equations are satisfied