A CHARACTERISTIC MIXED METHOD WITH DYNAMIC FINITE-ELEMENT SPACE FOR CONVECTION-DOMINATED DIFFUSION-PROBLEMS

A method for numerically solving convection-dominated diffusion problems using a combination of the method of characteristics with the mixed finite-element method is presented and analyzed in this paper. Different numbers of elements and different basis functions are implemented at different time levels in order to accurately resolve time-changing critical features. This new method provides the ability to obtain approximations of the solution and its gradient simultaneously, adopt large time steps, and make grid refinements and basis function adjustments at any time necessary. Convergence analysis and error estimates are established. Furthermore, comments are made on the efficiency and capability of the method for accurately solving transient problems with localized phenomena, such as fronts, shocks and boundary layers.