Peter Knabner

Alexander Jaust, Balthasar Reuter, Vadym Aizinger et al.

The third paper in our series on open source MATLAB / GNU Octave implementation of the discontinuous Galerkin (DG) method(s) focuses on a hybridized formulation. The main aim of this ongoing work is to develop rapid prototyping techniques covering a range of standard DG methodologies and suitable for small to medium sized applications. Our FESTUNG package relies on fully vectorized matrix / vector operations throughout, and all details of the implementation are fully documented. Once again, great care was taken to maintain a direct mapping between discretization terms and code routines as well as to ensure full compatibility to GNU Octave. The current work formulates a hybridized DG scheme for linear advection problem, describes hybrid approximation spaces on the mesh skeleton, and compares the performance of this discretization to the standard (element-based) DG method for different polynomial orders.

Andreas Rupp, Vadym Aizinger, Balthasar Reuter et al.

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 Galerkin method, and a statement of discrete energy stability is proved for the fully non-linear coupled system.

Peter Knabner, Gerhard Summ

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 element method with augmented Crouzeix-Raviart ansatz space. We discuss the reduction of unknowns by static condensation and propose Newton's method for the solution of local and global systems. Finally, we show, how such a nonlinear problem arises from the mixed formulation of Darcy-Forchheimer flow in porous media.

Peter Knabner, Gerhard Summ

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 by a cell-centered finite volume method. The resulting nonlinear equations are lineararized with Newton's method, the linearized systems are solved with a multigrid algorithm. Both subsystems are recoupled again in a Picard iteration. Numerical simulations based on a simplified model show how regions with different porosity stabilize the reaction zone inside the porous burner.

Peter Knabner, Gerhard Summ

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.

Matthias Herz, Peter Knabner

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 slow perikinetic coagulation. For the analytical investigation, we transform the model to a porous medium equation, which provides us access to the comprehensive analytical results for porous medium equations. Additionally, this transformation reveals a new application for porous medium equations. Eventually, we present numerical simulations of the model by solving the porous medium equation. We note that we solve the porous medium equation without any further regularization, which is often applied in this context.