Peter Knabner
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Error estimates for completely discrete FEM in energy‐type and weaker norms Open
The paper presents error estimates within a unified abstract framework for the analysis of FEM for boundary value problems with linear diffusion‐convection‐reaction equations and boundary conditions of mixed type. Since neither conformity …
Hybridizable discontinuous Galerkin method with mixed-order spaces for non-linear diffusion equations with internal jumps Open
We formulate a hybridizable discontinuous Galerkin method for parabolic equations with non-linear tensor-valued coefficients and jump conditions (Henry’s law). The analysis of the proposed scheme indicates the optimal convergence order for…
Local existence of strong solutions to micro–macro models for reactive transport in evolving porous media Open
Two-scale models pose a promising approach in simulating reactive flow and transport in evolving porous media. Classically, homogenised flow and transport equations are solved on the macroscopic scale, while effective parameters are obtain…
Error estimates for completely discrete FEM in energy-type and weaker norms Open
The paper presents error estimates within a unified abstract framework for the analysis of FEM for boundary value problems with linear diffusion-convection-reaction equations and boundary conditions of mixed type. Since neither conformity …
Global implicit solver for multiphase multicomponent flow in porous media with multiple gas components and general reactions Open
In order to study the efficiency of the various forms of trapping including mineral trapping scenarios for CO 2 storage behavior in deep layers of porous media, highly nonlinear coupled diffusion-advection-reaction partial differential equ…
Comparison study of phase-field and level-set method for three-phase systems including two minerals Open
We investigate reactive flow and transport in evolving porous media. Solute species that are transported within the fluid phase are taking part in mineral precipitation and dissolution reactions for two competing mineral phases. The evolut…
Local existence of strong solutions to micro-macro models for reactive transport in evolving porous media Open
Two-scale models pose a promising approach in simulating reactive flow and transport in evolving porous media. Classically, homogenized flow and transport equations are solved on the macroscopic scale, while effective parameters are obtain…
Homogenization of Two-Phase Flow in Porous Media From Pore to Darcy Scale: A Phase-Field Approach Open
We extend the two-scale expansion approach of periodic homogenization to\ninclude time scales and thus can tackle the full instationary\nNavier-Stokes-Cahn-Hilliard model at the pore scale as microscale. Time scale\nseparation allows us to…
Efficiency and Accuracy of Micro‐Macro Models for Mineral Dissolution Open
Micro‐macro models for dissolution processes are derived from detailed pore‐scale models applying upscaling techniques. They consist of flow and transport equations at the scale of the porous medium (macroscale). Both include averaged time…
Numerical benchmark study for flow in highly heterogeneous aquifers Open
Solving the flow problem is the first step in modeling contaminant transport in natural porous media formations. Since typical parameters for aquifers often lead to advection-dominated transport problems, accurate flow solutions are essent…
Numerical benchmark study for flow in highly heterogeneous aquifers Open
This article presents numerical investigations on accuracy and convergence\nproperties of several numerical approaches for simulating steady state flows in\nheterogeneous aquifers. Finite difference, finite element, discontinuous\nGalerkin…
Discontinuous Galerkin method for coupling hydrostatic free surface flows to saturated subsurface systems Open
We formulate a coupled surface/subsurface flow model that relies on hydrostatic equations with free surface in the free flow domain and on the Darcy model in the subsurface part. The model is discretized using the local discontinuous Galer…
Effective interface conditions for processes through thin heterogeneous layers with nonlinear transmission at the microscopic bulk-layer interface Open
In this paper, we consider a system of reaction-diffusion equations in a domain consisting of two bulk regions separated by a thin layer with thickness of order $ε$ and a periodic heterogeneous structure. The equations inside the layer dep…
Wavelet-based priors accelerate maximum-a-posteriori optimization in\n Bayesian inverse problems Open
Wavelet (Besov) priors are a promising way of reconstructing indirectly\nmeasured fields in a regularized manner. We demonstrate how wavelets can be\nused as a localized basis for reconstructing permeability fields with sharp\ninterfaces f…
Convergence analysis of a BDF2 / mixed finite element discretization of a Darcy–Nernst–Planck–Poisson system Open
This paper presents an a priori error analysis of a fully discrete scheme for the numerical solution of the transient, nonlinear Darcy–Nernst–Planck–Poisson system. The scheme uses the second order backward difference formula (BDF2) in tim…
Derivation of effective transmission conditions for domains separated by a membrane for different scaling of membrane diffusivity Open
We consider a system of non-linear reaction-diffusion equations in a domain consisting of two bulk regions separated by a thin layer with periodic structure. The thickness of the layer is of order $\epsilon$, and the equations inside the l…
Efficient Realization of the Mixed Finite Element Discretization for nonlinear Problems Open
We consider implementational aspects of the mixed finite element method for a special class of nonlinear problems. We establish the equivalence of the hybridized formulation of the mixed finite element method to a nonconforming finite elem…
Sovability of the mixed Formulation for Darcy-Forchheimer Flow in Porous Media Open
We consider the mixed formulation of the equations governing Darcy-Forchheimer flow in porous media. We prove existence and uniqueness of a solution for the stationary problem and the existence of a solution for the transient problem.
Hybrid Discretization Methods for Transient Numerical Simulation of\n Combustion in Porous Media Open
We present an algorithm for the numerical solution of the equations governing\ncombustion in porous inert media. The discretization of the flow problem is\nperformed by the mixed finite element method, the transport problems are\ndiscretiz…
Hybrid Discretization Methods for Transient Numerical Simulation of Combustion in Porous Media Open
We present an algorithm for the numerical solution of the equations governing combustion in porous inert media. The discretization of the flow problem is performed by the mixed finite element method, the transport problems are discretized …
Including van der Waals Forces in Diffusion-Convection Equations - Modeling, Analysis, and Numerical Simulations Open
This paper presents a model of van der Waals forces in the framework of diffusion-convection equations. The model consists of a nonlinear and degenerated diffusion-convection equation, which furthermore can be considered as a model for slo…
Modeling and simulation of coagulation according to DLVO-theory in a continuum model for electrolyte solutions Open
This paper presents a model of coagulation in electrolyte solutions. In this paper, the coagulation process is modeled according to DLVO-theory, which is an atomistic theory. On the other hand, we describe the dynamics in the electrolyte s…
Global existence of weak solutions of a model for electrolyte solutions - Part 2: Multicomponent case Open
This paper analytically investigates the Darcy-Poisson-Nernst-Planck system. This system is a mathematical model for electrolyte solutions. In this paper, we consider electrolyte solutions, which consist of a neutral fluid and multiple sus…