Variational multiscale a-posteriori error estimation for multi-dimensional transport problems

This paper presents an explicit a-posteriori error estimator for the multi-dimensional transport equation based on an approximation to the variational multiscale theory. The technique is adequate for methods with a local error distribution, such as stabilized methods, yielding efficiencies close to one for both, the hyperbolic and diffusive limit. Regarding the hyperbolic limit, previous work is extended to multi-dimensions, adequate norms are proposed for the computation of the error and proper error intrinsic time scales are calculated for the bilinear quadrilateral and the linear triangle. Furthermore, the model considers the element-interface error along the element edges, enabling the error prediction in the diffusive limit. The success of the method can be explained by the fact that in stabilized methods the element local problem captures most of the error and the proposed error intrinsic time scales are an approximation to the solution of the dual problem. The proposed technology, which can be implemented straightforwardly in existing codes, is extremely economical because it is a simple explicit postprocessing.