Adaptive Eulerian–Lagrangian finite element method for advection–dispersion

Abstract A new adaptive finite element method is proposed for the advection–dispersion equation using an Eulerian–Lagrangian formulation. The method is based on a decomposition of the concentration field into two parts, one advective and one dispersive, in a rigorous manner that does not leave room for ambiguity. The advective component of steep concentration fronts is tracked forward with the aid of moving particles clustered around each front. Away from such fronts the advection problem is handled by an efficient modified method of characteristics called single‐step reverse particle tracking. When a front dissipates with time, its forward tracking stops automatically and the corresponding cloud of particles is eliminated. The dispersion problem is solved by an unconventional Lagrangian finite element formulation on a fixed grid which involves only symmetric and diagonal matrices. Preliminary tests against analytical solutions of one‐ and two‐dimensional dispersion in a uniform steady‐state velocity field suggest that the proposed adaptive method can handle the entire range of Péclet numbers from 0 to ∞, with Courant numbers well in excess of 1.