GAMM2025

Session

S18: Numerical methods for differential equations

S18

8 Apr 2025, 08:30

Lecture and Conference Centre

Conveners

S18: Numerical methods for differential equations: S18.01

• Piotr Krzyżanowski

S18: Numerical methods for differential equations: S18.02

• Piotr Krzyżanowski

S18: Numerical methods for differential equations: S18.03

• Robert Altmann

S18: Numerical methods for differential equations: S18.04

• Roland Maier

S18: Numerical methods for differential equations: S18.05

• Roland Maier

S18: Numerical methods for differential equations: S18.06

• Robert Altmann

S18: Numerical methods for differential equations: S18.07

• Roland Maier

S18: Numerical methods for differential equations: S18.08

• Piotr Krzyżanowski

Description

For all fields of applications the mathematical models are primarily based on differential equations. Hence, their numerical solution plays a fundamental role in numerical mathematics. This section covers mainly the construction and the behavior of numerical methods for differential equations including those of ordinary as well as of partial differential type.

559. Lattice Boltzmann for 2D linear elastodynamics with Dirichlet and Neumann boundary conditions

Oliver Boolakee

08/04/2025, 08:30

Recent modeling frameworks to predict the mechanics of additive manufacturing processes involve both fluid and solid mechanics, with the former often described adopting the lattice Boltzmann method. Motivated by the wish to model all physics with the same method, we proposed a novel lattice Boltzmann formulation to solve the equations of linear elastostatics [1,2] and we now aim at extending...

560. Lattice Boltzmann Method for linear elastodynamics in 3D

Julie Weverbergh

08/04/2025, 08:50

The lattice Boltzmann method has recently been developed to solve problems in solid mechanics, starting with linear elastostatics [1, 2] and continuing with linear elastodynamics [3] in two dimensions. Based on these works, we propose an extension of the lattice Boltzmann method for linear elastodynamics in three dimensions. As in the 2D case, we transform the system into an equivalent...

561. Numerical solution of the fractional Euler-Bernoulli equation for a beam with fixed-supported and fixed-free ends

Anna Nowak

08/04/2025, 09:10

Fractional calculus is a powerful extension of the traditional calculus, which found its way to modeling complex phenomena such as anomalous diffusion, heat conduction, elasticity and plasticity, or micro/nano beams. The wide spectrum of applications of this branch of mathematical analysis is related to its nonlocal nature, which allows to describe problems where it is necessary to take into...

562. Numerical approximation of fractional compositions of differential operators with fixed memory length and its application to the problem of fractional continuum mechanics

Dominika Kustal

08/04/2025, 09:30

Fractional calculus is gaining increasing recognition among scientists from various fields of science, and one of its key applications is continuum mechanics. The introduction of nonlocality to models of this field has enabled the development of a fractional theory of continuum mechanics, which opens up new perspectives in modeling and analyzing the behavior of materials and structures. The...

563. Analysis of a Fractional-Order Model for Diabetes Mellitus Incorporating Education and Media Awareness Campaigns Using the Two-Step Newtonian Polynomial Approach

Vishalkumar Prajapati

08/04/2025, 09:50

Diabetes is rapidly emerging as a global epidemic, posing a significant threat to public health. Modeling the spread and management of diabetes is crucial for monitoring its growing prevalence and developing cost-effective strategies to mitigate its incidence and complications. This paper presents a fractional order nonlinear model for diabetes mellitus that incorporates the cumulative effect...

565. Fluid-Structure Interactions in ALE coordinates

Chiara Hergl

08/04/2025, 16:30

Biofilms describe a the colonization of bacteria on surfaces. They attract more and more bacteria and express a extracellular matrix (EPS). By that, a growing solid domain is formed. Biofilms mainly form in fluid domains. The bacteria are described as concentrations in our continuum description, where attraction and growth is described by an advection-diffusion-reaction equation (PDE) and the...

566. A monolithic space-time temporal multirate finite element framework for interface and volume coupled problems

Thomas Wick

08/04/2025, 16:50

In this presentation, we propose and computationally investigate a monolithic space–time multi-rate scheme for coupled problems. The novelty lies in the monolithic formulation of the multi-rate approach as this requires a careful design of the functional framework, corresponding discretization, and implementation. Our method of choice is a tensor-product Galerkin space–time discretization. The...

567. A Local hp Space-Time Multigrid Approach for Tensor-Product Finite Element Discretizations of the Stokes Equations

Nils Margenberg

08/04/2025, 17:10

We present a local hp space-time multigrid method for tensor-product finite element discretizations of the Stokes equations, using Qₚ/Pₚ₋₁ᵈⁱˢᶜ elements in space and a discontinuous Galerkin (DG) discretization in time. A key novelty of this work is the application of \textit{hp} multigrid techniques in both space and time. The method is facilitated by the matrix-free capabilities of the...

568. Space-time least-squares FEM for convection-diffusion problems

Christian Köthe

08/04/2025, 17:30

Instationary convection-diffusion problems arise in many applications, such as e.g., pollution simulations, heat transfer problems between thin domains, or in the modelling of flow and transport problems, to name but a few. In the advection-dominated case, the solutions are characterised by boundary layers, which lead to numerical instabilities and hence unphysical solutions when discretised...

569. Projection Methods in the Context of Nematic Crystal Flow

Maximilian Reiter

08/04/2025, 17:50

We present a continuous and a discontinuous linear Finite Element method based on a predictor-corrector scheme for the numerical approximation of the Ericksen-Leslie equations, a model for nematic liquid crystal flow including the non-convex unit-sphere constraint. As predictor step we propose a linear semi-implicit Finite Element discretization which naturally offers a local orthogonality...

570. Structure-preserving Model Reduction on Manifolds of port-Hamiltonian systems

Silke Glas

09/04/2025, 08:30

Capturing and preserving physical properties, e.g., system energy, stability and passivity, using data-driven methods is currently a highly-researched topic in surrogate modeling. To ensure that the desired physical properties are retained, structure-preserving projection techniques are used in the field in model reduction (MOR).

In this talk, we present structure-preserving MOR with...

571. Energy-preserving Arnoldi approximations for Gauss-Runge-Kutta integrators

Stefan Maier

09/04/2025, 09:10

This talk considers Poisson systems with a Hamiltonian describing the energy in the system. If the Hamiltonian is quadratic, the energy is exactly conserved using the Gauss-Legendre methods as numerical integrators. Since Gauss methods are implicit Runge-Kutta methods, a system of equations must be solved at each time step. The computation can be very expensive for high-dimensional systems...

572. Beyond 1D: A higher dimensional perspective on composite gas flow simulations in pipelines

Ashwin Sadanand Nayak

09/04/2025, 09:30

Accurate simulations of gas flow through pipeline networks can offer invaluable insights for network transmission and design operators. This becomes increasingly important with the integration of hydrogen-blended fuels in the development of hydrogen-based energy systems. Existing models primarily address single-component, one-dimensional flow within individual pipe elements, interconnected at...

573. Convergence of a Riemannian gradient method for the Gross-Pitaevskii energy functional in a rotating frame

Mahima Yadav

09/04/2025, 09:50

In this presentation, we focus on the numerical approximation of ground states of rotating Bose-Einstein condensates. This requires the minimization of the Gross-Pitaevskii energy functional on a Riemannian manifold. As an iterative solver for finding such minimizers we propose a generalized Riemannian gradient method with Sobolev gradients and an adaptively changing metric. We prove that the...

574. Reduced-order modeling and data assimilation with applications in structural health monitoring

Carmen Gräßle

09/04/2025, 16:30

In order to achieve realistic simulations of complex dynamics, a detailed mathematical model and a precise numerical method are required. However, there are often inaccuracies or terms in the model that are uncertain or unknown. This is where data assimilation comes in, which corrects the model with the help of measurement data. Since data assimilation typically involves numerous simulations...

575. Model Reduction for the Wave Equation beyond the limitations of the Kolmogorov N-width

Moritz Feuerle

09/04/2025, 17:10

The Reduced Basis Method (RBM) is a well-established model reduction technique to realize multi-query and/or realtime applications of parameterized partial differential equations (PPDEs). The RBM relies on a well-posed variational formulation of the PPDE under consideration. Since the RBM is a linear approximation method, the best possible rate of convergence is given by the Kolmogorov...

576. Fast Solution of the Wave Equation Using Model Order Reduction and the Laplace Transform

Fernando Henriquez

09/04/2025, 17:30

We present a novel, fast solver for the numerical approximation of linear, time-dependent partial differential equations, leveraging model order reduction techniques and the Laplace transform. Specifically, we consider the application of this method to the linear, second-order wave equation.

We begin by applying the Laplace transform to the evolution problem, which results in a...

577. Temperature Stratification in Lakes: Thermobaric Effects and Stability

Jonathan Irmscher

09/04/2025, 17:50

The stratification of lakes plays an important role for understanding and analysing the behaviour of lakes, as well as determining potential turning points of the system. In stably stratified lakes, the water layers can have drastically different chemical compositions, and an overturn may lead to dangerous reactions affecting the lake and surrounding environments.

Many common lake models...

578. Global Free Flight Optimization via Eikonal Approach

Arturas Jocas

09/04/2025, 18:10

The aviation sector is expanding annually, with an expected market size of nearly 340 billion USD in 2024. This growth raises significant concerns about fuel consumption and the corresponding CO2 emissions. Every day, approximately 1.3 million people commute within Europe using airplanes, resulting in roughly 240,000 tonnes of CO2 emitted into the atmosphere. Therefore, reducing fuel...

579. A posteriori error bounds without generic constants by the two-energies-principle

Dietrich Braess

10/04/2025, 08:30

Elliptic partial differential equations are often formulated as a variational problem, more precisely as a minimum problem for a suitable energy. A discretization by conforming finite elements provides an upper bound of the energy. Similarly the discretization of the dual variational problem yields a lower bound. The difference of the two energies provides an a posteriori error bound without...

580. Goal-oriented dual-weighted error estimation for first order Virtual Elements

Christian Sellmann

10/04/2025, 08:50

Goal-oriented error estimation is a powerful tool for efficient and reliable numerical simulations. It allows one to control the error for specific quantities of interest and thus, enables more reliable and efficient simulations. To compute the numerical solution, we utilize the Virtual Elements Method (VEM). The VEM can handle general polygonal meshes which allows for an easy incorporation of...

581. Error representations for goal-oriented a posteriori error estimation in elasto-plasticity with applications to mesh adaptivity

Arnold Tchomgue Simeu

10/04/2025, 09:10

In structural modeling, adaptive techniques based on a posteriori error estimation are well accepted to simulate finite element discretizations of partial differential equations. Based on a goal-oriented a posteriori error estimation, the quality of computational results is evaluated according to the physical significance of a specific quantity. In this contribution, several new methods of...

582. Quantum Realization of the Finite Element Method

Matthias Deiml

10/04/2025, 09:30

This talk presents a quantum algorithm for the solution of prototypical second-order linear elliptic partial differential equations discretized by d-linear finite elements on Cartesian grids of a bounded d-dimensional domain. An essential step in the construction is a BPX preconditioner, which transforms the linear system into a sufficiently well-conditioned one, making it amenable to quantum...

583. Minimal residual discretization of a class of fully nonlinear elliptic PDE

Ngoc Tien Tran

10/04/2025, 09:50

Solutions to partial differential equations (PDE) of second-order in nondivergence form are, in general, difficult to approximate by finite element methods due to the lack of a variational formulation. In such cases, minimal residual methods may be the method of choice due to their wide accessibility. The residual of this paper stems from the Alexandrov--Bakelman--Pucci maximum principle for...

584. Sparse low-rank approximation of multi-parametric partial differential equations

Huqing Yang

10/04/2025, 10:10

We consider a new approach to low-rank approximation of solutions to parameter -dependent PDEs that combines a representation in hierarchical tensor format with sparse polynomial expansions. We construct a low-rank adaptive Galerkin method for parametric elliptic problems that uses a tensor soft thresholding operation for rank reduction and discretization refinement based on lower-dimensional...

585. Optimal Order Pressure Trajectory Approximation for Stokes Systems: Set of Pressure Solutions and its Post-Processing

Markus Bause

10/04/2025, 14:00

We address the optimal order approximation of the pressure trajectory for an equal-order in time variational discretization of velocity and pressure in Stokes systems. In the literature, the pressure approximation of Stokes and Navier-Stokes systems has attracted less attention than the velocity approximation, even though being of equal importance for applications, for instance, for the...

586. Stabilized finite elements for incompressible Navier-Stokes flows on manifolds

Michael Wolfgang Kaiser

10/04/2025, 14:20

We present residual-based stabilization techniques for finite element methods to simulate Navier-Stokes flows in stress-divergence form on curved surfaces. The mixed formulation in velocity and pressure variables leads to the inf-sup condition. One may use Taylor-Hood elements, e.g., [1,2], or equal-order element pairs for velocity and pressure together with a stabilization scheme, e.g., the...

587. Mixed finite element for the Stokes eigenvalue problem

Tugay Dagli

10/04/2025, 14:40

In this contribution, we present a novel three-field finite element for the Stokes eigenvalue problem. To this end, we approximate the Hellinger-Reissner (mixed) formulation where the symmetry of the stress tensor σ is dealt with in a weak form by introducing a Lagrange multiplier that represents the conservation of angular momentum. We consider the space of tensors whose rows consist of an...

588. A positivity preserving scheme for a coupled Chemotaxis--(Navier--)Stokes system

Christos Pervolianakis

10/04/2025, 15:00

In this talk, we consider a coupled Chemotaxis--(Navier--)Stokes system on a bounded domain Ω ⊂ ℝ². The system consists of two parabolic partial differential equations coupled with an incompressible (Navier--)Stokes equation, describing the functions (z, c, 𝐮), where z = z(𝐱, t) denotes the cell density, c = c(𝐱, t) the substance concentration, 𝐮 = 𝐮(𝐱, t) with 𝐮 = (u₁, u₂)ᵀ the velocity field...

589. Goal-Oriented Adaptivity Techniques for Convection-Dominated Transport and Flow Problems

Marius Paul Bruchhäuser

10/04/2025, 15:20

The Dual Weighted Residual (DWR) Method has attracted researchers’ interest in many fields of application problems since it was introduced by Becker and Rannacher at the turn of the last millennium. With regard to an efficient numerical approximation of the underlying model problem, the DWR approach yields a posteriori error estimator measured in goal quantities of physical interest, that can be...

590. Analysis and numerics of nonlinear PDE systems in porous media flow models

Simon Boisserée

10/04/2025, 15:40

In models of porous media flows, the porosity of the solid matrix is often treated as a static quantity. However, under certain circumstances, such as in soft sedimentary rocks or in magma flows, the porosity of the solid material can evolve under the influence of fluid pressure which can lead to the formation of solitary porosity waves and of higher-porosity channels. We consider a system of...

591. Accelerating exponential integrators

Alexander Ostermann

10/04/2025, 16:30

Exponential integrators are a class of time integration methods used to solve large systems of evolution equations. They integrate the linear part of the problem (almost) exactly and use an explicit scheme for the nonlinearity. This makes them highly accurate and stable, especially when the nonlinearity is weak. These integrators are widely used in stiff problems of semilinear parabolic...

592. Semi-explizit Discretization of Thermo-poroelasticity

Jochewed Schmeck

10/04/2025, 16:50

We present an approach to solve the equations of thermo-poroelasticity. This system describes the dynamics of an elastic solid, which contains pores which are filled with a liquid. The model gives a coupled system that consists of one elliptic equation, describing the elastic deformation of the solid under physical stress and two parabolic equations, for the flow and the temperature of the...

593. A rigorous (validated) method for numerically solving systems of Delay Differential Equations

Robert Szczelina

10/04/2025, 17:10

The Delay Differential Equations (DDEs) are the class of differential equations where the derivative of the solution at the current time depends not only on the state of the system at that time, but also on the states in the past. This formulation leads to a (semi)dynamical system on an infinite dimensional phase space, similar to what happens in Partial Differential Equations (PDEs). Thus,...

594. Parameter-robust unfitted finite element methods for a Maxwell interface problem

Tim Haubold

11/04/2025, 08:30

Geometrically unfitted finite element methods such as CutFEM, Finite Cell, XFEM or unfitted DG methods have been developed and applied successfully in the last decades to a variety of problems, ranging from scalar PDEs on stationary domains to systems of PDEs on moving domains and PDEs on level set surfaces. These approaches combined with established tools of finite element methods allowed to...

595. A comparative study of H(curl) and Lagrange based interpolations for the magnetic field

Maximilian Vorwerk

11/04/2025, 08:50

Permanent magnets, such as neodymium-iron-boron (NdFeB), play a crucial role in enhancing the efficiency of power conversion devices, including wind turbines, sensors, and electric motors, cf. [1]. Due to their enormous potential to address current technological and societal challenges, such as reducing CO2 emissions, they are the subject of intensive research. The primary goals include...

596. On the accuracy of the boundary element method for problems with discontinuous geometries

Michał Paweł Rajski

11/04/2025, 09:10

The boundary element method (BEM) is one of the very interesting domain based approximation techniques that has been successfully applied to solve numerous physical problems. It solves a given partial differential equation (PDE) using only boundary discretization, which significantly alleviates the meshing effort, but also limits the possible PDEs to be solved to only linear ones. Furthermore,...

597. A methodology for calculating rotor-stator flows based on finite volume mesh-tying

Kian Karimian

11/04/2025, 09:30

Mesh-tying is a common computational approach for solving transport equations on two disjoint subdomains whose associated grids do not conform at the common interface. In the present contribution, we propose a finite volume adaptation of this general concept that is optimized for block-structured projection-based flow solvers. A characteristic trait of the method is that the interfacial mass...

598. Numerical methods for nonlocal and nonlinear parabolic equations with applications

Łukasz Płociniczak

11/04/2025, 09:50

Many natural and industrial phenomena exhibit nonlocal behaviour in temporal or spatial dimensions. The former is responsible for processes for which its whole history influences the present state. The latter, on the other hand, indicates that faraway regions of the domain may have some impact on local points. This is useful in describing media of high heterogeneity.

Partial differential...

564. Investigation of a Fuzzy Fractional Diabetes Model Involving Glucose–Insulin Alliance Scheme with a Fuzzy Double Parametric Approach

Parthkumar P. Sartanpara