Robert Altmann

Robert Altmann, Attila Karsai, Philipp Schulze

We investigate discretization strategies for a recently introduced class of energy-based models. The model class encompasses classical port-Hamiltonian systems, generalized gradient flows, and certain systems with algebraic constraints. Our framework combines existing ideas from the literature and systematically addresses temporal discretization, spatial discretization, and model order reduction, ensuring that all resulting schemes are dissipation-preserving in the sense of a discrete dissipation inequality. For this, we use a Petrov--Galerkin ansatz together with appropriate projections. Numerical results for a nonlinear circuit model, the Cahn--Hilliard equation, and a doubly nonlinear parabolic equation illustrate the effectiveness of the approach.

Robert Altmann, Eric Chung, Roland Maier et al.

We consider a strongly heterogeneous medium saturated by an incompressible viscous fluid as it appears in geomechanical modeling. This poroelasticity problem suffers from rapidly oscillating material parameters, which calls for a thorough numerical treatment. In this paper, we propose a method based on the local orthogonal decomposition technique and motivated by a similar approach used for linear thermoelasticity. Therein, local corrector problems are constructed in line with the static equations, whereas we propose to consider the full system. This allows to benefit from the given saddle point structure and results in two decoupled corrector problems for the displacement and the pressure. We prove the optimal first-order convergence of this method and verify the result by numerical experiments.

Shubin Fu, Robert Altmann, Eric T. Chung et al.

In this work, we employ the Constraint Energy Minimizing Generalized Multiscale Finite Element Method (CEM-GMsFEM) to solve the problem of linear heterogeneous poroelasticity with coefficients of high contrast. The proposed method makes use of the idea of energy minimization with suitable constraints in order to generate efficient basis functions for the displacement and the pressure. These basis functions are constructed by solving a class of local auxiliary optimization problems based on eigenfunctions containing local information on the heterogeneity. Techniques of oversampling are adapted to enhance the computational performance. Convergence of first order is shown and illustrated by a number of numerical tests.

Robert Altmann, Christoph Zimmer

We consider partial differential equations on networks with a small parameter $ε$, which are hyperbolic for $ε>0$ and parabolic for $ε=0$. With a combination of an $ε$-expansion and Runge-Kutta schemes for constrained systems of parabolic type, we derive a new class of time discretization schemes for hyperbolic systems on networks, which are constrained due to interconnection conditions. For the analysis we consider the coupled system equations as partial differential-algebraic equations based on the variational formulation of the problem. We discuss well-posedness of the resulting systems and estimate the error caused by the $ε$-expansion.

Robert Altmann, Abdullah Mujahid, Benjamin Unger

We study the construction and convergence of semi-explicit and iterative decoupling schemes for an elliptic-parabolic problem using higher-order Runge-Kutta methods. For the semi-explicit schemes, which are constructed using a nearby delay system with $k$ time delays, we establish the convergence of $k$th-order Runge-Kutta methods under a weak coupling condition. We develop the convergence analysis by adapting the Fourier stability and perturbation techniques of [Lubich, Ostermann, Math. Comp., 64(210):601--627, 1995]. The key tool is the generating function framework, in which the Runge-Kutta discretization is encoded through an operator-valued function. Stability estimates are then obtained via Parseval's identity on the unit circle. We further present convergence results for iterative (fixed-stress and undrained-split) higher-order Runge-Kutta schemes. Here, a spectral decomposition of the Schur complement operator is central. Finally, we provide numerical examples to verify the proven convergence results.

Robert Altmann, Abdullah Mujahid, Benjamin Unger

For the iterative decoupling of elliptic-parabolic problems such as poroelasticity, we introduce time discretization schemes up to order $5$ based on the backward differentiation formulae. Its analysis combines techniques known from fixed-point iterations with the convergence analysis of the temporal discretization. As main result, we show that the convergence depends on the interplay between the time step size and the parameters for the contraction of the iterative scheme. Moreover, this connection is quantified explicitly, which allows for balancing the single error components. Several numerical experiments illustrate and validate the theoretical results, including a three-dimensional example from biomechanics.

Robert Altmann

The weak formulation of parabolic problems with dynamic boundary conditions is rewritten in form of a partial differential-algebraic equation. More precisely, we consider two dynamic equations with a coupling condition on the boundary. This constraint is included explicitly as an additional equation and incorporated with the help of a Lagrange multiplier. Well-posedness of the formulation is shown.

Robert Altmann, Alexander Ostermann

We consider Lie and Strang splitting for the time integration of constrained partial differential equations with a nonlinear reaction term. Since such systems are known to be sensitive with respect to perturbations, the splitting procedure seems promising as we can treat the nonlinearity separately. This has some computational advantages, since we only have to solve a linear constrained system and a nonlinear ODE. However, Strang splitting suffers from order reduction which limits its efficiency. This is caused by the fact that the nonlinear subsystem produces inconsistent initial values for the constrained subsystem. The incorporation of an additional correction term resolves this problem without increasing the computational cost. Numerical examples including a coupled mechanical system illustrate the proven convergence results.