On Adaptive Eulerian---Lagrangian Method for Linear Convection---Diffusion Problems

Authors:
Xiaozhe Hu;Young-Ju Lee;Jinchao Xu;Chen-Song Zhang
Affiliations:
Department of Engineering Mechanics, Kunming University of Science and Technology, Kunming, China and Department of Mathematics, The Pennsylvania State University, University Park, USA 16802;Department of the Mathematics, Rutgers, The State University of New Jersey, Piscataway, USA 08854;Department of Mathematics, The Pennsylvania State University, University Park, USA 16802;NCMIS and LSEC, Academy of Mathematics and System Sciences, Beijing, China 100190
Venue:
Journal of Scientific Computing
Year:
2014

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Abstract

In this paper, we consider the adaptive Eulerian---Lagrangian method (ELM) for linear convection---diffusion problems. Unlike classical a posteriori error estimations, we estimate the temporal error along the characteristics and derive a new a posteriori error bound for ELM semi-discretization. With the help of this proposed error bound, we are able to show the optimal convergence rate of ELM for solutions with minimal regularity. Furthermore, by combining this error bound with a standard residual-type estimator for the spatial error, we obtain a posteriori error estimators for a fully discrete scheme. We present numerical tests to demonstrate the efficiency and robustness of our adaptive algorithm.