TY - INPR A1 - Egger, Herbert T1 - A mixed variational discretization for non-isothermal compressible flow in pipelines N2 - 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. Y1 - 2017 ER - TY - INPR A1 - Egger, Herbert A1 - Radu, Bogdan T1 - Super-convergence and post-processing for mixed finite element approximations of the wave equation N2 - 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. Y1 - 2017 ER - TY - INPR A1 - Egger, Herbert A1 - Böttcher, Anke T1 - Energy stable discretization of Allen-Cahn type problems modeling the motion of phase boundaries N2 - 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. Y1 - 2017 ER - TY - INPR A1 - Egger, Herbert A1 - Kugler, Thomas A1 - Liljegren-Sailer, Björn A1 - Marheineke, Nicole A1 - Mehrmann, Volker T1 - On structure preserving model reduction for damped wave propagation in transport networks N2 - 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 construction 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. KW - partial differential-algebraic equations KW - port-Hamiltonian systems KW - Galerkin projection Y1 - 2017 ER - TY - INPR A1 - Egger, Herbert A1 - Kugler, Thomas T1 - Uniform exponential stability of Galerkin approximations for damped wave systems N2 - 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 stabil- ity 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 esti- mates under minimal regularity assumptions. We also prove unconditional and uniform exponential stability for the time discretization by certain one-step methods. The valid- ity of the theoretical results as well as the necessity of some of the conditions required for our analysis are demonstrated in numerical tests Y1 - 2016 ER - TY - INPR A1 - Egger, Herbert A1 - Kugler, Thomas T1 - Damped wave systems on networks: exponential stability and uniform approximations N2 - 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 realiza- tion 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. Y1 - 2016 ER - TY - JOUR A1 - Egger, Herbert A1 - Kugler, Thomas A1 - Strogies, Nikolai T1 - Parameter identification in a semilinear hyperbolic system JF - Inverse Problems N2 - 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 con- traction 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 varia- tional regularization method is considered for its stable solution. We prove existence of minimizers for the Tikhonov functional and discuss the convergence of the regularized so- lutions 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 Y1 - 2016 VL - 33 IS - 055022 ER - TY - JOUR A1 - Egger, Herbert T1 - A robust conservative mixed finite element method for compressible flow on pipe networks N2 - We consider the numerical approximation of compressible flow in a pipe net- work. 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 fur- ther 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 demon- stration of the robustness and accuracy of the new method Y1 - 2016 ER - TY - INPR A1 - Egger, Herbert A1 - Giesselmann, Jan T1 - Regularity and long time behavior of a doubly nonlinear parabolic problem and its discretization N2 - We study a doubly nonlinear parabolic problem arising in the modeling of gas transport in pipelines. Using convexity arguments and relative entropy estimates we show uniform bounds and exponential stability of discrete approximations obtained by a finite element method and implicit time stepping. Due to convergence of the approximations to weak solutions of the problem, our results also imply regularity, uniqueness, and long time stability of weak solutions of the continuous problem. KW - gas transport KW - doubly nonlinear parabolic problems KW - relative entropy estimates KW - exponential stability KW - structure preserving discretization Y1 - 2023 ER -