Stabilization Techniques for the Finite Element Method Applied to Advection Dominated Problems

Abstract

This thesis in the mathematical field of numerics of partial differential equations deals with different stabilization techniques of finite element discretizations of advection-dominated problems. The underlying advection–diffusion equation is used, e.g., in hydrogeology, where it describes the transport of groundwater-dissolved matter through the porous soil matrix at an averaged scale. The transport is caused by the actual groundwater flow (advection) as well as by Brownian particle motion and grain-structure-related dispersion (diffusion). Discretizing the advection–diffusion equation in space using the classical finite element method (FEM), unphysical oscillations occur if advection dominates diffusion. The quotient of these two quantities defines the Peclet number. For high Peclet numbers, the numerical solution also attains negative concentration values. Three different stabilization techniques are considered in this thesis: The first one uses a finite volume discretization for the advective part. For equations of hyperbolic character, finite volume methods are known to have better properties than the FEM. The second method is the streamline diffusion method (streamline upwind Petrov–Galerkin, SUPG) for polynomial degrees one and two. Finally, a not yet widely-established method is considered: Algebraic flux correction (AFC). AFC limits the mass flux between degrees of freedom such that a given prescribed principle is preserved. For example, one can guarantee the discrete non-negativity of concentrations using the AFC method. This is desirable as negative concentrations are physically not meaningful. AFC is formulated on the algebraic level, not on the variational one, and introduces additional non-linearities. Those can be handled by a fixed-point iteration or an appropriate linearization. In any case, AFC turns out to have the highest computational costs compared to the other two stabilization methods. While one can apply SUGP for arbitrary polynomial degrees, the finite volume stabilization and AFC in the standard linear mass lumping version can only be applied using first-order polynomials. Show more