Adaptive Discontinuous Galerkin Methods for Non-linear Reactive Flows

Production of drinkable water for human consumption and for other supply purposes, as well as exploitation of available geo-resources by human intervention, are the main characteristics of the geological subsurface. Protection and sustainable management of water resources is one of the key problems in environmental engineering. Modeling and forecasting of soil and ground water contamination in industrial areas and in large scale agricultural land pose new challenges in geosciences. In this aspect, accurate modeling and simulation of coupled ground and surface water flows is a necessity. Chemically reactive components such as dissolved minerals, colloids, or contaminants are transported by advection and diffusion over long distances through some highly heterogeneous porous media.

The discontinuous Galerkin (dG) method was introduced by Reed and Hill [73] in 1973 for steady-state neutron transport as an hyperbolic problem. This was followed by other studies; by Bassi and Rebay [12] for the compressible Navier-Stokes equations, Cockburn and Shu [31] developed the local discontinuous Galerkin (ldG) method for advection-diffusion equations, and Peraire and Persson [69] introduced the compact discontinuous Galerkin (cdG) method. Independent of the dG methods, interior penalty (IP) methods have been developed for elliptic and parabolic problems by Douglas and Dupont [40] and Wheeler [95]. Then, in the 1980’s, Arnold et al. [6] proposed a unified classification and analysis of various kinds of dG methods. Later on, the dG methods were developed for elliptic problems [8, 24, 77] and for problems with advection [7, 14, 51, 56].

In this chapter we investigate and apply adaptive dG algorithms for the stationary semi-linear ADR equations of the model (1.1). We give the existence and uniqueness results of the elliptic system. The main focus of this chapter is handling of unphysical oscillations at the interior/boundary layers in advection dominated problems resulting through the discretization in space by applying an adaptive algorithm using residual-based robust a posteriori error estimates for the stationary model. The results obtained in this chapter for the stationary model will be a key ingredient in the follow-up chapter for the non-stationary models. Since the stiffness matrices obtained by dG methods become more dense and ill-conditioned with increasing order of dG polynomials, the resulting linear system of equations have to preconditioned. For this reason, we introduce in this chapter the matrix reordering and iterative partitioning technique in [86]. We give the details of the construction of the matrix reordering and partitioning technique, and demonstrate its efficiency numerically.

Application of adaptive dG methods and a posteriori error estimates to problems in geoscience are reviewed recently in [33]. Most of the applications of dG methods in geoscience concern reactive transport with advection [13, 62, 84] and strong permeability contrasts such as layered reservoirs [90] or vanishing and varying diffusivity posing challenges in computations [72]. The permeability in heterogeneous porous and fractured media varies over orders of magnitude in space, which results in highly variable flow field, where the local transport is dominated by advection or diffusion [85]. Accurate and efficient numerical solution of ADR equations to predict macroscopic mixing, anomalous transport of solutes and contaminants for a wide range of parameters like permeability and Péclet numbers, different flow velocities and reaction rates and reaction rates are challenging problems [85]. In order to resolve the complex flow patterns accurately, higher order time stepping methods like exponential time stepping methods are used [85].

In this book adaptive algorithms are developed for efficient discretization of advection dominated stationary and non-stationary semi-linear ADR equations. In order to handle the unphysical oscillations due to the advection, we have applied a symmetric interior penalty Galerkin (SIPG) method as an alternative to the well-known stabilized continuous FEM methods such as the streamlined upwind Petrov-Galerkin (SUPG) method. We have given a detailed construction of SIPG formulation on the general Poisson equation, and we have discussed the effect of the penalty parameter in Chapter 2. In Chapter 3, we gave existence and uniqueness results for stationary semi-linear ADR equations. We have shown that the space-time adaptive algorithm is robust and can resolve not only the layers produced by advection but also the sharp fronts due to the non-linear reaction as an alternate to the shock/discontinuity capturing techniques in the literature.We have also shown that adaptive dG approximations for stationary problems are more accurate than the Galerkin least squares FEMs and shock/discontinuity capturing techniques. Moreover, we have introduced an efficient iterative method, matrix reordering technique, as a preconditioner to solve the linear systems arising from the Newton’s method applied to the discrete system of stationary semi-linear ADR equations.