Error analysis for characteristics-based methods for degenerate parabolic problems

We consider characteristics-based finite element methods for solving nonlinear, degenerate, advection-diffusion equations. These equations have applications in the simulation of petroleum reservoirs and groundwater aquifers and in the modeling of free boundary problems. Standard finite element Galerkin methods have been studied for these equations. In this paper, we analyze the characteristics-based finite element methods for them. The main difficulty in the analysis is that the equations are degenerate and the solution lacks regularity. Here we develop a technique that respects the degeneracy and the known minimal regularity This technique is based on the Green operator for standard elliptic equations and is developed directly for the degenerate advection-diffusion equations. We concentrate our analysis on the modified method of characteristics (MMOC) and one of its variants, the modified method of characteristics with adjusted advection (MMOCAA), which conserves mass. We derive error estimates in various norms. The extension to other variants is discussed. The present technique is also applied to nondegenerate problems; error estimates previously obtained for the MMOC are derived under much weaker regularity assumptions on the solution, and the error estimates for the MMOCAA appear new even in the nondegenerate case. Finally, numerical results are presented to show the sharpness of the error estimates derived.