Herbert Egger

Herbert Egger, Thomas Kugler, Björn Liljegren-Sailer et al.

We consider the discretization and subsequent model reduction of a system of partial differential-algebraic equations describing the propagation of pressure waves in a pipeline network. Important properties like conservation of mass, dissipation of energy, passivity, existence of steady states, and exponential stability can be preserved by an appropriate semi-discretization in space via a mixed finite element method and also during the further dimension reduction by structure preserving Galerkin projection which is the main focus of this paper. Krylov subspace methods are employed for the construciton of the reduced models and we discuss modifications needed to satisfy certain algebraic compatibility conditions; these are required to ensure the well-posedness of the reduced models and the preservation of the key properties. Our analysis is based on the underlying infinite dimensional problem and its Galerkin approximations. The proposed algorithms therefore have a direct interpretation in function spaces; in principle, they are even applicable directly to the original system of partial differential-algebraic equations while the intermediate discretization by finite elements is only required for the actual computations. The performance of the proposed methods is illustrated with numerical tests and the necessity for the compatibility conditions is demonstrated by examples.

Herbert Egger, Bernd Hofmann

Conditional stability estimates allow us to characterize the degree of ill-posedness of many inverse problems, but without further assumptions they are not sufficient for the stable solution in the presence of data perturbations. We here consider the stable solution of nonlinear inverse problems satisfying a conditional stability estimate by Tikhonov regularization in Hilbert scales. Order optimal convergence rates are established for a-priori and a-posteriori parameter choice strategies. The role of a hidden source condition is investigated and the relation to previous results for regularization in Hilbert scales is elaborated. The applicability of the results is discussed for some model problems, and the theoretical results are illustrated by numerical tests.

Herbert Egger, Thomas Kugler

We consider a damped linear hyperbolic system modelling the propagation of pressure waves in a network of pipes. Well-posedness is established via semi-group theory and the existence of a unique steady state is proven in the absence of driving forces. Under mild assumptions on the network topology and the model parameters, we show exponential stability and convergence to equilibrium. This generalizes related results for single pipes and multi-dimensional domains to the network context. Our proof of the exponential stability estimate is based on a variational formulation of the problem, some graph theoretic results, and appropriate energy estimates. The main arguments are rather generic and can be applied also for the analysis of Galerkin approximations. Uniform exponential stability can be guaranteed for the resulting semi-discretizations under mild compatibility conditions on the approximation spaces. A particular realization by mixed finite elements is discussed and the theoretical results are illustrated by numerical tests in which also bounds for the decay rate are investigated.

Herbert Egger, Klemens Fellner, Jan-Frederik Pietschmann et al.

We consider the numerical solution of coupled volume-surface reaction-diffusion systems having a detailed balance equilibrium. Based on the conservation of mass, an appropriate quadratic entropy functional is identified and an entropy-entropy dissipation inequality is proven. This allows us to show exponential convergence to equilibrium by the entropy method. We then investigate the discretization of the system by a finite element method and an implicit time stepping scheme including the domain approximation by polyhedral meshes. Mass conservation and exponential convergence to equilibrium are established on the discrete level by arguments similar to those on the continuous level and we obtain estimates of optimal order for the discretization error which hold uniformly in time. Some numerical tests are presented to illustrate these theoretical results. The analysis and the numerical approximation are discussed in detail for a simple model problem. The basic arguments however apply also in a more general context. This is demonstrated by investigation of a particular volume-surface reaction-diffusion system arising as a mathematical model for asymmetric stem cell division.

Herbert Egger

We present a general abstract framework for the systematic numerical approximation of dissipative evolution problems. The approach is based on rewriting the evolution problem in a particular form that complies with an underlying energy or entropy structure. Based on the variational characterization of smooth solutions, we are then able to show that the approximation by Galerkin methods in space and discontinuous Galerkin methods in time automatically leads to numerical schemes that inherit the dissipative behavior of the evolution problem. The proposed framework is rather general and can be applied to a wide range of applications. This is demonstrated by a detailed discussion of a variety examples ranging from diffusive partial differential equations to Hamiltonian and gradient systems.

Herbert Egger, Matthias Schlottbom

The numerical solution of time-dependent radiative transfer problems is challenging, both, due to the high dimension as well as the anisotropic structure of the underlying integro-partial differential equation. In this paper we propose a general framework for designing numerical methods for time-dependent radiative transfer based on a Galerkin discretization in space and angle combined with appropriate time stepping schemes. This allows us to systematically incorporate boundary conditions and to preserve basic properties like exponential stability and decay to equilibrium also on the discrete level. We present the basic a-priori error analysis and provide abstract error estimates that cover a wide class of methods. The starting point for our considerations is to rewrite the radiative transfer problem as a system of evolution equations which has a similar structure like first order hyperbolic systems in acoustics or electrodynamics. This analogy allows us to generalize the main arguments of the numerical analysis for such applications to the radiative transfer problem under investigation. We also discuss a particular discretization scheme based on a truncated spherical harmonic expansion in angle, a finite element discretization in space, and the implicit Euler method in time. The performance of the resulting mixed PN-finite element time stepping scheme is demonstrated by computational results.

Herbert Egger, Vsevolod Shashkov, Kersten Schmidt

We consider the efficient numerical solution of coupled dynamical systems, consisting of a small nonlinear part and a large linear time invariant part, possibly stemming from spatial discretization of an underlying partial differential equation. The linear subsystem can be eliminated in frequency domain and for the numerical solution of the resulting integro-differential algebraic equations, we propose a a combination of Runge-Kutta or multistep time stepping methods with appropriate convolution quadrature to handle the integral terms. The resulting methods are shown to be algebraically equivalent to a Runge-Kutta or multistep solution of the coupled system and thus automatically inherit the corresponding stability and accuracy properties. After a computationally expensive pre-processing step, the online simulation can, however, be performed at essentially the same cost as solving only the small nonlinear subsystem. The proposed method is, therefore, particularly attractive, if repeated simulation of the coupled dynamical system is required.

Herbert Egger, Lucas Schöbel-Kröhn

We consider the Keller-Segel model of chemotaxis on one-dimensional networks. Using a variational characterization of solutions, positivity preservation, conservation of mass, and energy estimates, we establish global existence of weak solutions and uniform bounds. This extends related results of Osaki and Yagi to the network context. We then analyze the discretization of the system by finite elements and an implicit time-stepping scheme. Mass lumping and upwinding are used to guarantee the positivity of the solutions on the discrete level. This allows us to deduce uniform bounds for the numerical approximations and to establish order optimal convergence of the discrete approximations to the continuous solution without artificial smoothness requirements. In addition, we prove convergence rates under reasonable assumptions. Some numerical tests are presented to illustrate the theoretical results.

Herbert Egger, Thomas Kugler

We consider the numerical approximation of linear damped wave systems by Galerkin approximations in space and appropriate time-stepping schemes. Based on a dissipation estimate for a modified energy, we prove exponential decay of the physical energy on the continuous level provided that the damping is effective everywhere in the domain. The methods of proof allow us to analyze also a class of Galerkin approximations based on a mixed variational formulation of the problem. Uniform exponential stability can be guaranteed for these approximations under a general compatibility condition on the discretization spaces. As a particular example, we discuss the discretization by mixed finite element methods for which we obtain convergence and uniform error estimates under minimal regularity assumptions. We also prove unconditional and uniform exponential stability for the time discretization by certain one-step methods. The validity of the theoretical results as well as the necessity of some of the conditions required for our analysis are demonstrated in numerical tests.

Herbert Egger, Thomas Kugler, Björn Liljegren-Sailer

We consider the discretization of a semilinear damped wave equation arising, for instance, in the modeling of gas transport in pipeline networks. For time invariant boundary data, the solutions of the problem are shown to converge exponentially fast to steady states. We further prove that this decay behavior is inherited uniformly by a class of Galerkin approximations, including finite element, spectral, and structure preserving model reduction methods. These theoretical findings are illustrated by numerical tests.

Herbert Egger, Thomas Kugler, Nikolai Strogies

We consider the identification of a nonlinear friction law in a one-dimensional damped wave equation from additional boundary measurements. Well-posedness of the governing semilinear hyperbolic system is established via semigroup theory and contraction arguments. We then investigte the inverse problem of recovering the unknown nonlinear damping law from additional boundary measurements of the pressure drop along the pipe. This coefficient inverse problem is shown to be ill-posed and a variational regularization method is considered for its stable solution. We prove existence of minimizers for the Tikhonov functional and discuss the convergence of the regularized solutions under an approximate source condition. The meaning of this condition and some arguments for its validity are discussed in detail and numerical results are presented for illustration of the theoretical findings.

Herbert Egger, Christoph Erath, Robert Schorr

We consider the numerical approximation of parabolic-elliptic interface problems by the non-symmetric coupling method of MacCamy and Suri [Quart. Appl. Math., 44 (1987), pp. 675--690]. We establish well-posedness of this formulation for problems with non-smooth interfaces and prove quasi-optimality for a class of conforming Galerkin approximations in space. Therefore, error estimates with optimal order can be deduced for the semi-discretization in space by appropriate finite and boundary elements. Moreover, we investigate the subsequent discretization in time by a variant of the implicit Euler method. As for the semi-discretization, we establish well-posedness and quasi-optimality for the fully discrete scheme under minimal regularity assumptions on the solution. Error estimates with optimal order follow again directly. Our analysis is based on estimates in appropriate energy norms. Thus, we do not use duality arguments and corresponding estimates for an elliptic projection which are not available for the non-symmetric coupling method. Additionally, we provide again error estimates under minimal regularity assumptions. Some numerical examples illustrate our theoretical results.

Herbert Egger

We consider the numerical approximation of compressible flow in a pipe network. Appropriate coupling conditions are formulated that allow us to derive a variational characterization of solutions and to prove global balance laws for the conservation of mass and energy on the whole network. This variational principle, which is the basis of our further investigations, is amenable to a conforming Galerkin approximation by mixed finite elements. The resulting semi-discrete problems are well-posed and automatically inherit the global conservation laws for mass and energy from the continuous level. We also consider the subsequent discretization in time by a problem adapted implicit time stepping scheme which leads to conservation of mass and a slight dissipation of energy of the full discretization. The well-posedness of the fully discrete scheme is established and a fixed-point iteration is proposed for the solution of the nonlinear systems arising in every single time step. Some computational results are presented for illustration of our theoretical findings and for demonstration of the robustness and accuracy of the new method.

Herbert Egger, Thomas Kugler

We consider a parameter dependent family of damped hyperbolic equations with interesting limit behavior: the system approaches steady states exponentially fast and for parameter to zero the solutions converge to that of a parabolic limit problem. We establish sharp estimates and elaborate their dependence on the model parameters. For the numerical approximation we then consider a mixed finite element method in space together with a Runge-Kutta method in time. Due to the variational and dissipative nature of this approximation, the limit behavior of the infinite dimensional level is inherited almost automatically by the discrete problems. The resulting numerical method thus is asymptotic preserving in the parabolic limit and uniformly exponentially stable. These results are further shown to be independent of the discretization parameters. Numerical tests are presented for a simple model problem which illustrate that the derived estimates are sharp in general.

Herbert Egger

A general framework for the numerical approximation of evolution problems is presented that allows to preserve exactly an underlying Hamiltonian- or gradient structure. The approach relies on rewriting the evolution problem in a particular form that complies with the underlying geometric structure. The Galerkin approximation of a corresponding variational formulation in space then automatically preserves this structure which allows to deduce important properties for appropriate discretization schemes including projection based model order reduction. We further show that the underlying structure is preserved also under time discretization by a Petrov-Galerkin approach. The presented framework is rather general and allows the numerical approximation of a wide range of applications, including nonlinear partial differential equations and port-Hamiltonian systems. Some examples will be discussed for illustration of our theoretical results and connections to other discretization approaches will be revealed.

Herbert Egger, Bogdan Radu

We consider the numerical approximation of single phase flow in porous media by a mixed finite element method with mass lumping. Our work extends previous results of Wheeler and Yotov, who showed that mass lumping together with an appropriate choice of basis allows to eliminate the flux variables locally and to reduced the mixed problem in this way to a finite volume discretization for the pressure only. Here we construct second order approximations for hybrid meshes in two and three space dimensions which, similar to the method of Wheeler and Yotov, allows the local elimination of the flux variables. A full convergence analysis of the method is given for which new arguments and, in part, also new quadrature rules and finite elements are required. Computational tests are presented for illustration of the theoretical results.

Herbert Egger

We consider the non-isothermal flow of a compressible fluid through pipes. Starting from the full set of Euler equations, we propose a variational characterization of solutions that encodes the conservation of mass, energy, and entropy in a very direct manner. This variational principle is suitable for a conforming Galerkin approximation in space which automatically inherits the basic physical conservation laws. Three different spaces are used for approximation of density, mass flux, and temperature, and we consider a mixed finite element method as one possible choice of suitable approximation spaces. We also investigate the subsequent discretization in time by a problem adapted implicit time stepping scheme for which exact conservation of mass as well as a slight dissipation of energy and increase of entropy are proven which are due to the numerical dissipation of the implicit time discretization. The main arguments of our analysis are rather general and allow us to extend the approach with minor modification to more general boundary conditions and flow models taking into account friction, viscosity, heat conduction, and heat exchange with the surrounding medium.

Anke Böttcher, Herbert Egger

We study the systematic numerical approximation of a class of Allen-Cahn type problems modeling the motion of phase interfaces. The common feature of these models is an underlying gradient flow structure which gives rise to a decay of an associated energy functional along solution trajectories. We first study the discretization in space by a conforming Galerkin approximation of a variational principle which characterizes smooth solutions of the problem. Well-posedness of the resulting semi-discretization is established and the energy decay along discrete solution trajectories is proven. A problem adapted implicit time-stepping scheme is then proposed and we establish its well-posed and decay of the free energy for the fully discrete scheme. Some details about the numerical realization by finite elements are discussed, in particular the iterative solution of the nonlinear problems arising in every time-step. The theoretical results are illustrated by numerical tests which also provide further evidence for asymptotic expansions of the interface velocities derived by Alber et al.

Herbert Egger, Thomas Kugler, Vsevolod Shashkov

This paper studies the discretization of gas transport in pipeline networks by an inexact Petrov-Galerkin method. A full convergence analysis is presented for single pipes under the assumption of a linear friction law and the possible extension to pipe networks is discussed. The generalization to nonlinear gas transport models and the efficient implementation by hybridization is investigated numerically.

Herbert Egger, Matthias Schlottbom

We consider the numerical approximation of boundary conditions in radiative transfer problems by a perfectly matched layer approach. The main idea is to extend the computational domain by an absorbing layer and to use an appropriate reflection boundary condition at the boundary of the extended domain. A careful analysis shows that the consistency error introduced by this approach can be made arbitrarily small by increasing the size of the extension domain or the magnitude of the artificial absorption in the surrounding layer. A particular choice of the reflection boundary condition allows us to circumvent the half-space integrals that arise in the variational treatment of the original vacuum boundary conditions and which destroy the sparse coupling observed in numerical approximation schemes based on truncated spherical harmonics expansions. A combination of the perfectly matched layer approach with a mixed variational formulation and a PN-finite element approximation leads to discretization schemes with optimal sparsity pattern and provable quasi-optimal convergence properties. As demonstrated in numerical tests these methods are accurate and very efficient for radiative transfer in the scattering regime.

Herbert Egger, Gabriel Teschner

We consider the stable reconstruction of flow geometry and wall shear stress from measurements obtained by magnetic resonance imaging. As noted in a review article by Petersson, most approaches considered so far in the literature seem not be satisfactory. We therefore propose a systematic reconstruction procedure that allows to obtain stable estimates of flow geometry and wall shear stress and we are able to quantify the reconstruction errors in terms of bounds for the measurement errors under reasonable smoothness assumptions. A full analysis of the approach is given in the framework of regularization methods. In addition, we discuss the efficient implementation of our method and we demonstrate its viability, accuracy, and regularizing properties for experimental data.

Herbert Egger, Bogdan Radu

A novel mass-lumping strategy for a mixed finite element approximation of Maxwell's equations is proposed. On structured orthogonal grids the resulting method coincides with the spatial discretization of the Yee scheme. The proposed method, however, generalizes naturally to unstructured grids and anisotropic materials and thus yields a variational extension of the Yee scheme for these situations.

Herbert Egger, Bogdan Radu

We consider the numerical approximation of acoustic wave propagation problems by mixed BDM(k+1)-P(k) finite elements on unstructured meshes. Optimal convergence of the discrete velocity and super-convergence of the pressure by one order are established. Based on these results, we propose a post-processing strategy that allows us to construct an improved pressure approximation from the numerical solution. Corresponding results are well-known for mixed finite element approximations of elliptic problems and we extend these analyses here to the hyperbolic problem under consideration. We also consider the subsequent time discretization by the Crank-Nicolson method and show that the analysis and the post-processing strategy can be generalized to the fully discrete schemes. Our proofs do not rely on duality arguments or inverse inequalities and the results therefore apply also for non-convex domains and non-uniform meshes.

Herbert Egger

We consider the discretization of parabolic initial boundary value problems by finite element methods in space and a Runge-Kutta time stepping scheme. Order optimal a-priori error estimates are derived in an energy-norm under natural smoothness assumptions on the solution and without artificial regularity conditions on the parameters and the domain. The key steps in our analysis are the careful treatment of time derivatives in the H(-1)-norm and the the use of an L2-projection in the error splitting instead of the Ritz projector. This allows us to restore the optimality of the estimates with respect to smoothness assumptions on the solution and to avoid artificial regularity requirements for the problem, usually needed for the analysis of the Ritz projector, which limit the applicability of previous work. The wider applicability of our results is illustrated for two irregular problems, for which previous results can either not by applied or yield highly sub-optimal estimates.

Herbert Egger, Fritz Kretzschmar, Sascha M. Schnepp et al.

The modeling and simulation of electromagnetic wave propagation is often accompanied by a restriction to bounded domains which requires the introduction of artificial boundaries. The corresponding boundary conditions should be chosen in order to minimize parasitic reflections. In this paper, we investigate a new type of transparent boundary condition for a discontinuous Galerkin Trefftz finite element method. The choice of a particular basis consisting of polynomial plane waves allows us to split the electromagnetic field into components with a well specified direction of propagation. The reflections at the artificial boundaries are then reduced by penalizing components of the field incoming into the space-time domain of interest. We formally introduce this concept, discuss its realization within the discontinuous Galerkin framework, and demonstrate the performance of the resulting approximations by numerical tests. A comparison with first order absorbing boundary conditions, that are frequently used in practice, is made. For a proper choice of basis functions, we observe spectral convergence in our numerical test and an overall dissipative behavior for which we also give some theoretical explanation.

Herbert Egger, Fritz Kretzschmar, Sascha M. Schnepp et al.

We consider the discretization of electromagnetic wave propagation problems by a discontinuous Galerkin Method based on Trefftz polynomials. This method fits into an abstract framework for space-time discontinuous Galerkin methods for which we can prove consistency, stability, and energy dissipation without the need to completely specify the approximation spaces in detail. Any method of such a general form results in an implicit time-stepping scheme with some basic stability properties. For the local approximation on each space-time element, we then consider Trefftz polynomials, i.e., the subspace of polynomials that satisfy Maxwell's equations exactly on the respective element. We present an explicit construction of a basis for the local Trefftz spaces in two and three dimensions and summarize some of their basic properties. Using local properties of the Trefftz polynomials, we can establish the well-posedness of the resulting discontinuous Galerkin Trefftz method. Consistency, stability, and energy dissipation then follow immediately from the results about the abstract framework. The method proposed in this paper therefore shares many of the advantages of more standard discontinuous Galerkin methods, while at the same time, it yields a substantial reduction in the number of degrees of freedom and the cost for assembling. These benefits and the spectral convergence of the scheme are demonstrated in numerical tests.