Application of Variational a-Posteriori Multiscale Error Estimation to Higher-Order Elements

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 methods, 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.