A coupled finite element-global random walk approach to advection-dominated transport in porous media with random hydraulic conductivity - Suciu, Nicolae; Radu, Adrian Florin; Prechtel, Alexander; Brunner, Fabian; Knabner, Peter (Journal of Computational and Applied Mathematics, 2013)
Solute transport through heterogeneous porous media considered in environmental and industrial problems is often characterized by high Péclet numbers. In this paper we develop a new numerical approach to advection-dominated transport consisting of coupling an accurate mass-conservative mixed finite element method (MFEM), used to solve Darcy flows, with a particle method, stable and free of numerical diffusion, for non-reactive transport simulations. The latter is the efficient global random walk (GRW) algorithm, which performs the simultaneous tracking of arbitrarily large collections of particles on regular lattices at computational costs comparable to those of single-trajectory simulations using traditional particle tracking (PT). MFEM saturated flow solutions are computed for spatially heterogeneous hydraulic conductivities parameterized as samples of random fields. The coupling is achieved by projecting the velocity field from the MFEM basis onto the regular GRW lattice. Preliminary results show that MFEM-GRW is tens of times faster than the full MFEM flow and transport simulation. © 201
Keywords: Advection-dominated transport; Porous media; Random walk methods

A global implicit solver for miscible reactive multiphase multicomponent flow in porous media - Brunner, Fabian; Knabner, Peter (Computational Geosciences, 2019)
We present a numerical framework for efficiently simulating partially miscible two-phase flow with multicomponent reactive transport in porous media using the global implicit approach. The mathematical model consists of coupled and nonlinear partial differential equations, ordinary differential equations, and algebraic equations. Our approach is based on a model-preserving reformulation using the reduction scheme of Krautle and Knabner (Water Resour. Res. 43(3), 2007), Hoffmann et al. (Comput. Geosci. 16(4):1081-1099, 2012) to transform the system. Moreover, a nonlinear, implicitly defined resolution function to reduce its size is employed. By choosing persistent primary variables and using a complementarity approach, mineral reactions and the local appearance and disappearance of the gas phase can be handled without a discontinuous switch of primary variables. In each time step of the Euler-implicit time stepping scheme, the discrete nonlinear systems are solved using the Semismooth Newton method for linearization using the global implicit approach. Thus, we obtain an efficient, robust, and stable simulation method allowing for large time steps and avoiding the potential drawbacks of splitting approaches.
CRIS-Team WoS Importer:2019-03-12

Analysis of a Modified Second-Order Mixed Hybrid BDM_1 Finite Element Method for Transport Problems in Divergence Form - Brunner, Fabian; Fischer, Julian; Knabner, Peter (SIAM Journal on Numerical Analysis, 2016)
We prove optimal second order convergence of a modified lowest-order Brezzi-Douglas-Marini (BDM1) mixed finite element scheme for advection-diffusion problems in divergence form. If advection is present, it is known that the total flux is approximated only with first-order accuracy by the classical BDM1 mixed method, which is suboptimal since the same order of convergence is obtained if the computationally less expensive Raviart-Thomas (RT0) element is used. The modification that was first proposed by Brunner et al. [Adv. Water Res., 35 (2012), pp. 163-171] is based on the hybrid problem formulation and consists in using the Lagrange multipliers for the discretization of the advective term instead of the cellwise constant approximation of the scalar unknown.
Keywords: advection-diffusion problem; mixed finite element methods; suboptimal convergence; optimal convergence

Analysis of an upwind-mixed hybrid finite element method for transport problems - Brunner, Fabian; Radu, Adrian Florin; Knabner, Peter (SIAM Journal on Numerical Analysis, 2014)
We prove optimal order convergence of an upwind-mixed hybrid finite element scheme for linear parabolic advection-diffusion-reaction problems. It was introduced in [Radu et al., Adv. Water Resources, 34(2011), pp. 47-61] and is based on an Euler-implicit mixed hybrid finite element discretization of the problem in fully mass conservative form using the Raviart-Thomas mixed finite element of lowest order on triangular meshes. Optimal order convergence in time and space is obtained for the fully discrete formulation. The scheme provides the same order of convergence as the standard upwind-mixed method, while it is more efficient since a local elimination of variables is possible with our choice of the upwind weights. The theoretical findings are sustained by a numerical experiment. © 2014 Society for Industrial and Applied Mathematic
Keywords: A priori error estimates; Advection-diffusion problem; Advection-dominance; Mixed finite element method; Upwind weighting

An Improved Optimal Order Mixed Finite Element Method for Semilinear Transport Problems - Bause, Markus; Brunner, Fabian; Knabner, Peter; Radu, Adrian Florin (Numerical Mathematics and Advanced Applications 2011, 2012)
We propose and study the numerical approximation of an advection-diffusion-reaction model equation by a modified Brezzi–Douglas–Marini mixed finite element method.Nonlinear advection is admitted, arising in complex and coupled flow and transport systems.In contrast to the classical variant of this approach, optimal second-order convergence of the scalar and the vector variable is ensured.No loss of rate of convergence due to the presence of the advection term is observe

FV upwind stabilization of FE discretizations for advection-diffusion problems - Brunner, Fabian; Frank, Florian; Knabner, Peter (Finite Volumes for Complex Applications VII-Methods and Theoretical Aspects, 2014)
We apply a novel upwind stabilization of a mixed hybrid finite element method of lowest order to advection–diffusion problemswith dominant advection and compare it with a finite element scheme stabilized by finite volume upwinding. Both schemes are locally mass conservative and employ an upwind-weighting formula in the discretization of the advective term. Numerical experiments indicate that the upwind-mixed method is competitive with the finite volume method. It prevents the appearance of spurious oscillations and produces nonnegative solutions for strongly advection-dominated problems, while the amount of artificial diffusion is lower than that of the finite volume method. This makes the method attractive for applications in which too much numerical diffusion is critical and may lead to false predictions; e.g., if highly nonlinear reactive processes take place only in thin interaction region

Optimal order convergence of a modified BDM 1 mixed finite element scheme for reactive transport in porous media - Brunner, Fabian; Radu, Adrian Florin; Bause, Markus; Knabner, Peter (Advances in Water Resources, 2012)
We present a mass conservative finite element approach of second order accuracy for the numerical approximation of reactive solute transport in porous media modeled by a coupled system of advection-diffusion-reaction equations. The lowest order Brezzi-Douglas-Marini (BDM ) mixed finite element method is used. A modification based on the hybrid form of the approach is suggested for the discretization of the advective term. It is demonstrated numerically that this leads to optimal second order convergence of the flux variable. The modification improves the convergence behavior of the classical BDM scheme, which is known to be suboptimal of first order accuracy only for advection-diffusion problems; cf. [8]. Moreover, the new scheme shows more robustness for high Péclet numbers than the classical approach. A comparison with the Raviart-Thomas element (RT ) of second order accuracy for the approximation of the flux variable is also presented. For the case of strongly advection-dominated problems we propose a full upwind scheme. Various numerical studies, including also a nonlinear test problem, are presented to illustrate the numerical performance properties of the considered numerical methods. © 2011 Elsevier Lt
Keywords: Mixed finite element methods; Optimal order convergence; Reactive transport