Reliable and efficient error control for an adaptive Galerkin-characteristic method for convection-dominated diffusion problems

Abstract

An efficient and reliable a-posteriori error estimator is developed for a characteristic-Galerkin finite element method for time-dependent convection-dominated problems. An adaptive algorithm with variable time and space steps is proposed and studied. At each time step in this algorithm grid coarsening occurs solely at the final iteration of the adaptive procedure, meaning that only time and space refinement is allowed before the final Communicated by Aihui Zhou. Ming Cui is supported in part by a State Scholarship Fund of China and the Independent 320 M. Cui et al.iteration. It is proved that at each time step this adaptive algorithm is capable of reducing errors below a given tolerance in a finite number of iteration steps. Numerical results are presented to check the theoretical analysis.

The lowest order characteristic mixed finite element scheme for convection-dominated diffusion problem

“…Solving such convection-dominated diffusion problems numerically is challenging, the standard finite difference method (FDM) or finite element method (FEM) does not perform well as it often exhibits sharp fronts and excessive nonphysical oscillations. In order to overcome these difficulties, many effective schemes have been proposed for this equation, such as streamline diffusion method [2,3], Eulerian-Lagrangian method [4,5], least-squares-mixed FEM [6,7], the modified characteristics-Galerkin FEM [8,9], adaptive Galerkin-characteristic method [10], characteristic nonconforming FEM [11][12][13][14], characteristic MFEM [15,16] and expanded characteristic MFEM [1,17], characteristic FDM [18,19], characteristic-finite volume element method [20,21] and characteristic-mixed covolume methods [22], and the integral equation formulations based on boundary element method [23,24]. The modified characteristic Galerkin FEM is a combination of characteristics method and standard Galerkin FEM.…”