Doktorandenseminar des WIAS
FG3: Numerische Mathematik und Wissenschaftliches
Rechnen /
RG3: Numerical Mathematics
and Scientific Computing/
LG5: Numerik für innovative
Halbleiter-Bauteile
LG5: Numerics for
innovative semiconductor devices
Seminar Numerische Mathematik / Numerical mathematics seminars
aktuelles Programm / current program
Archiv
Hybrid-/Online-Vorträge finden über ''Zoom'' statt. Der Zoom-Link wird jeweils ca. 15 Minuten vor Beginn des Gesprächs versendet. / The zoom link will be send about 15 minutes before the start of the talk. People who are not members of the research group 3 and who are interested in participating should contact christian.merdon@wias-berlin.de to obtain the zoom login details.
Donnerstag, 05. 06. 2025, 14:00 Uhr (WIAS-405/406)
Adrian Hill (Technische Universität Berlin)
SparseConnectivityTracer.jl
Donnerstag, 22. 05. 2025, 15:30 Uhr (WIAS-405/406)
Holger Stephan (WIAS Berlin)
Improvement of the diffusion equation using hyperbolic systems and memory equationsThe diffusion equation is the simplest equation to describe the motion of a particle in a random medium. It is the 0th moment (mass balance) of the more exact kinetic equation (Kramers equation). Higher momentum approximations such as the 1st momentum (momentum balance) and the 2nd momentum (energy balance) are also common. Moment approximations must be closed. We calculate the optimal approximations for various higher moment systems for the case of no external potential. This leads to hyperbolic systems which, as is well known, are difficult to solve numerically. However, systems of diffusion equations with memory terms can be derived from them without loss of information. These can also be converted into common systems of diffusion equations also without loss of information using a recently developed method, which can be solved numerically very well. The result is a much more precise description of the diffusion process.
Donnerstag, 15. 05. 2025, 14:00 Uhr (WIAS-405/406)
Peter Munch (Technische Universität Berlin)
The deal.II finite-element library: recent developments and applicationsdeal.II is an open-source general-purpose finite-element C++ library with origins at the University of Heidelberg in 1997. Since then, it has evolved into a mature and worldwide developed and used mathematical software library (including at WIAS and TU Berlin). It is the basis of applications ranging from traditional application fields like computational solid mechanics, fluid mechanics, biomechanics, and geosciences to more “exotic” fields of research such as quantum and plasma physics. The scalability of deal.II has been shown by solving partial differential equations with trillions of unknowns on 300k processes. For its outstanding contributions to the development and use of mathematical and computational tools and methods for the solution of science and engineering problems, deal.II was awarded the prestigious “SIAM/ACM Prize in Computational Science and Engineering 2025”. In this talk, we will give a short introduction into deal.II, discuss recent developments in the library and present new use cases.
Donnerstag, 17. 04. 2025, 14:00 Uhr (WIAS-405/406)
Marwa Zainelabdeen (WIAS Berlin)
Gradient-robust finite element-finite volume scheme for the compressible Stokes equationsWe consider a steady compressible Stokes problem on a domain Ω ⊂ Rd, where d ∈ {2, 3}, in primitive variables velocity, pressure and non-constant density (u, p, ϱ). A barotropic flow is assumed, where the pressure depends solely on the density un- der an exponential equation of state p = cM ϱγ for γ ≥ 1. A finite element scheme for the momentum balance, coupled to a finite volume discretization for the continuity equation ∇ · (ϱu) = 0, was proposed in [1] for a linear equation of state (γ = 1). In this talk, we present an extension of the scheme to the nonlinear equation of state (γ > 1). The scheme satisfies several desired structural properties, namely stability, convergence, the preservation of non-negativity and mass constraints for the density, and gradient-robustness. The latter property is related to the locking phenomenon observed in incompressible flow at high Reynolds number regimes, which carries over to the compressible setting. To achieve gradient-robustness, we employed the reconstruction operator proposed in [2]. The structural properties of the scheme were tested using various numerical benchmark problems.
Donnerstag, 27. 03. 2025, 14:00 Uhr (WIAS-405/406)
Kees Vuik (Delft University of Technology)
Resolving divergence: the first multigrid scheme for the highly indefinite Helmholtz equation using classical componentsIn this talk, we (V. Dwarka and C. Vuik) present the first stand-alone classical multigrid solver for the highly in- definite 2D Helmholtz equation with constant costs per iteration, addressing a longstanding open problem in numerical analysis [1]. Our work covers both large constant and non- constant wavenumbers up to k = 500 in 2D. We obtain a full V − and W −cycle without any level-dependent restrictions. Another powerful feature is that it can be combined with the computationally cheap weighted Jacobi smoother. The key novelty lies in the use of higher-order inter-grid transfer operators [2]. When combined with coarsening on the Complex Shifted Laplacian, rather than the original Helmholtz operator, our solver is h−independent and scales linearly with the wavenumber k. If we use GMRES(3) smoothing we obtain k− independent convergence, and can coarsen on the original Helmholtz operator, as long as the higher-order transfer operators are used. This work opens doors to study robustness of contemporary solvers, such as machine learning solvers inspired by multigrid components, without adding to the black-box complexity.
Donnerstag, 13. 03. 2025, 14:00 Uhr (WIAS-405/406)
John Papadopoulos (WIAS Berlin)
Hierarchical proximal Galerkin: a fast hp-FEM solver for variational problems with pointwise inequality constraintsIn 2024 a novel algorithm, called the latent variable proximal point method (LVPP), was proposed for solving PDE-constrained optimization problems with pointwise inequality constraints. It is formulated on the infinite-dimensional level and enjoys many of the favourable characteristics of a semismooth Newton method including mesh independence and provable fast convergence rates. Moreover, no special considerations are required for its discretization allowing one to use any number of techniques such as finite volume, spectral methods, PINNs, or, as we will explore in this talk, hp-finite element methods (hp-FEM).
We will explore the use of LVPP for the obstacle problem (an obstacle-type constraint), the generalized elastic-plastic torsion problem (a gradient-type constraint), and the thermoforming problem (an obstacle-type quasi-variational inequality). We will consider discretizations up to pre-tensor degree p=82, which far exceed any previously reported discretizations for such problems. This is achieved by discretizing the nonlinear saddle point subproblems arising in the LVPP solver with a hierarchical p-FEM discretization, employing a Newton linearization, and solving the Newton systems via GMRES coupled with a block preconditioner. We will see that the number of the Newton iterations to reach convergence are hp-independent and the number of preconditioned GMRES iterations only grow mildly with p/h. We will directly compare the wall-clock solve times with low-order discretizations and observe that high-order discretizations often attain a target error significantly faster, achieving up to a 100 times speedup.
Donnerstag, 27. 02. 2025, 14:00 Uhr (WIAS-405/406)
El-Houssaine Quenjel (La Rochelle Université)
Stable finite volume methods for transient convection-diffusion systems with anisotropyThis talk is divided into two parts: positive/weakly monotone schemes and Hybrid-Upwinding discretizations of diphasic flows in heterogeneous porous media. First of all, I will begin with a short introduction to motivate the necessity of a positive finite volume approximation, especially in the context of diffusion problems or porous media flow systems. Next, I will review the basics for the construction and analysis of such a discretization with a focus on stability. Then, an ap- plication to a standard control volume finite element (CVFE) method will be investigated. As the CVFE approach is not positive (monotone), I will show how to introduce nonlinear corrections in the fluxes without changing the method’s stencil. As a result, the discrete maximum principle can be ensured, but accuracy can be deteriorated in case of problems with strong anisotropy of the diffusion or permeability matrix. To cope with this inconvenience, an advanced flux correction is taken in place to eliminate the anti-diffusion terms, only when they occur. This technique, referred to as weak monotony, enables the preservation of some piecewise linear bounds, which is sometimes more important than positivity. Each part is accompanied by numerical examples to illustrate and compare the efficiency and robustness of the corresponding scheme.
In the second part, I will present a Hybrid-Upwinding discretization of the incompressible two- phase flow model with discontinuous capillary forces. The idea makes use total velocity formulation and key approximation of the fluxes to ensure stability and therefore the main ingredient for the convergence analysis of the numerical scheme. Some numerical experiment will be exposed to show the efficiency and the robustness of the HU-scheme, especially in terms of the nonlinear convergence of the Newton-Rphason solver, compared to the pioneer phase-by-phase upstream discretization.
Dienstag, 14. 01. 2025, 13:30 Uhr (WIAS-405/406)
Jinchao Xu (King Abdullah University of Science and Technology)
Finite neuron methods for partial differential equationsIn this talk, I will report several theoretical and numerical studies of finite neuron method for partial differential equations. Topics include approximation properties of neural network functions, error estimates for the finite neuron method, convergence analysis of training algorithms, the relationship between ReLU^k finite neuron method with the tradition finite element and spectral method, and the comparison of their efficiency for high dimensional problems.
Dienstag, 14. 01. 2025, 15:00 Uhr (WIAS-405/406)
Christopher Rackauckas (Massachusetts Institute of Technology, Julia Lab)
The numerical analysis of scientific machine learning and differentiable programming
Donnerstag, 28. 11. 2024, 14:00 Uhr (HVP11A-3.13)
Lucas. O. Muller (University of Trento)
Towards real-life computional haemodynamicsUnderstanding the fundamental forces driving human physiology relies on a deep understanding of the circulation. During our daily life, the cardiovascular system is exposed to a wide range of biomechanical environments. This complexity requires a systemic approach to integrate phenomena across various temporal and spatial scales. Most of the research efforts in this do- main have focused on simplified cardiovascular scenarios such as the resting state, the supine position, and occasionally disease conditions. However, the cardiovascular system experiences large deviations from such prototypical conditions, which are, in many cases, highly beneficial to sustaining a healthy life. The impact of fundamental factors such as respiration, control mech- anisms, gravity, diverse physiological states (e.g., exercise or sleep conditions), and the integration of organ-specific physiology and disease into blood flow models has only been partially addressed. This talk aims to explore ongoing work in the field of computational hemodynamics at the systemic scale exploiting an anatomically detailed arterio-venous network 1D model [1] and highlighting modelling and numerical challenges related to simulations that intend to incorporate three aspects that are inextri- cably coupled to the circulation in our routine life: (i) the effect of respiration, (ii) the effect of orthostatic stress, and (iii) the presence of massive microvascular networks.
Donnerstag, 24. 10. 2024, 14:00 Uhr (WIAS-405/406)
Winnifried Wollner (Universität Hamburg)
Gradient-robustness in the context of optimal control
Dienstag, 01. 10. 2024, 13:30 Uhr (WIAS-405/406)
Philip L. Lederer (Universität Hamburg)
High-order projection-based upwind method for implicit large eddy simulationWe assess the ability of high-order (hybrid) discontinuous Galerkin methods to simulate under-resolved turbulent flows. The capabilities of the mass conserving mixed stress-yielding method as structure resolving large eddy simulation (LES) solver are examined. A comparison of a variational multiscale model to no-model or an implicit model approach is presented via numerical results. In addition, we present a novel approach for turbulent modeling in wall-bounded flows which can be interpreted as an extension of the classical variational multiscale idea to implicit LES via discontinuous Galerkin methods. This new technique called high-order projection-based upwind (HOPU) technique provides a more accurate representa- tion of the actual subgrid scales in the near wall region and gives promising results for highly under-resolved flow problems. We consider the turbulent channel flow and periodic hill flow problem as well as a flow over an Eppler airfoil.
Mittwoch, 18. 09. 2024, 10:00 Uhr (WIAS-405/406)
Albert Pool (DLR e.V.)
Nonlinear dynamics as a ground-state solution on quantum computersIn this talk, we describe how variational quantum algorithms (VQAs) can be used to solve nonlinear partial differential equations (PDEs) with applications in battery or semiconductor research, as well as other engineering problems. VQAs are hybrid quantum-classical algorithms, where a cost function is calculated on a quantum computer and its parameters are optimised classically. We present a spacetime representation, inspired by the Feynman–Kitaev Hamiltonian, where the solution to a PDE at all times is obtained by minimising just one function.
We describe how quantum nonlinear processing units can be used to evaluate the required cost function efficiently. We propose an adaptive strategy to mitigate the barren plateau problem, which means that gradients vanish exponentially with the number of qubits.
The approach is illustrated for the nonlinear Burgers equation. We optimise the problem on a noiseless simulator and run the circuits containing the result on IBM Q System One and Quantinuum Model H1, demonstrating that current quantum computers are capable of accurately reproducing the exact results.
Albert J. Pool, Alejandro D. Somoza, Conor Mc Keever, Michael Lubasch, Birger Horstmann
Dienstag, 16. 07. 2024, 13:30 Uhr (WIAS-ESH)
Nishant Ranwan (IISER Thiruvananthapuram)
Existence of a Weak Solution to the Fluid-Structure Interaction Problem of Blood Flow in Coronary ArteryThis work is dedicated to proving the existence of a weak solution to the nonlinear coupled fluid-structure interaction (FSI) problem arising in modeling the blood flow through the coronary artery. Considering blood as a homogeneous, incompressible, Newtonian fluid, the $3D$ Navier--Stokes equations model the flow. We adopt the St. Venant--Kirchhoff elasticity model for the arterial wall. With the help of dynamic and kinematic boundary conditions, we define the fluid-structure coupling (two-way) at the interface. Using the operator splitting approach, we split the coupled problem into two parts: the structure sub-problem (SSP) in the Galerkin framework and the piece-wise stationary fluid sub-problem (FSP). In order to establish the existence of a weak solution, the initial step is to tackle the SSP. Following this, the fluid domain is updated through the Harmonic Extension of the structural displacement. Subsequently, the focus shifts to solving the coupled Navier--Stokes equations within the updated fluid domain.
Donnerstag, 06. 06. 2024, 14:00 Uhr (WIAS-ESH)
Steffen Maass (WIAS/TU Berlin)
Symbolic-numeric modeling of electrochemical interfaces
Donnerstag, 06. 06. 2024, 14:30 Uhr (WIAS-ESH)
Tim Siebert (WIAS/TU Berlin)
Algorithmic differentiation with a focus on higher-order derivatives
Dienstag, 04. 06. 2024, 13:30 Uhr (WIAS-ESH)
Jürgen Fuhrmann (WIAS)
The nonlinear finite Volume solver VoronoiFVM.jl: capabilities, applications and plans
Dienstag, 04. 06. 2024, 14:00 Uhr (WIAS-ESH)
Daniel Runge (WIAS)
Mass-conservative reduced basis approach for heterogeneous catalysis
Dienstag, 28. 05. 2024, 13:30 Uhr (WIAS-406)
Sarah Katz (WIAS)
Pulsatile flow in the aorta: modelling and simulationThe modelling and simulation of blood flow is one of the major applications of (computational) fluid dynamics in medicine. We present a brief overview of some past results on turbulence and viscosity modelling for aortic flow and introduce currently ongoing work on ML-accelerated simulations.
Dienstag, 28. 05. 2024, 14:00 Uhr (WIAS-406)
Francesco Romor (WIAS)
Registration-based data assimilation of aortic blood flowsReal-world applications often require the execution of expensive numerical simulations on different geometrical domains, represented by computational meshes. This is especially true when performing shape optimization or when these domains model shapes from the real world, such as anatomical organs of the human body. How we can develop surrogate models or perform data assimilation in these contexts is the main concern of this work. Even if a rich database is acquired for a specific application, sometimes it cannot be exploited completely because each quantity of interest is supported on its own computational mesh. In fact, the employment of neural network architectures tailored for this framework is still not consolidated in the literature. A solution is represented by shape registration: a reference template geometry is designed from a cohort of available geometries which can then be mapped onto it. Through registration, a correspondence between every geometry of the database of parametric PDEs’ solutions is designed.
Donnerstag, 23. 05. 2024, 14:00 Uhr (WIAS-ESH)
Alfonso Caiazzo (WIAS)
Multiscale modeling of elastic materials & Overview of applications in the assimilation of biomedical dataThe first part of the talk will describe our recent work concerning the modeling of biphasic tissues composed of an elastic matrix (2D or 3D) and fluid inclusions of co-dimension 2 (0D or 1D, respectively). The model is based on a finite element immersed method where the effect of the fluid is accounted via proper singular terms. The dimensionality reduction is achieved using a reduced Lagrange multipliers formulation of the resulting PDE. The second part of the talk is dedicated to an overview of applications (past, present, and upcoming) in quantitative biomedicine focusing on the role of mathematical modeling and scientific computing for data assimilation problems.
Dienstag, 21. 05. 2024, 13:30 Uhr (WIAS-ESH)
Christian Merdon (WIAS)
Pressure-robustness in Navier--Stokes simulations
Donnerstag, 16.05.2024, 14:00 Uhr (WIAS-ESH)
Medine Demir (WIAS)
Pressure-robust approximation of the incompressible Navier--Stokes equations in a rotating frame of referenceA pressure-robust space discretization of the incompressible Navier--Stokes equations in a rotating frame of reference is considered. The discretization employs divergence-free, $H^1$-conforming mixed finite element methods like Scott--Vogelius pairs. An error estimate for the velocity is derived that tracks the dependency of the error bound on the coefficients of the problem, in particular on the angular velocity. Numerical examples support the theoretical results.
Dienstag, 14. 05. 2024, 13:30 Uhr (WIAS-ESH)
Cristian Cárcamo (WIAS)
Equal-order finite element methods for coupled and multiphysics problemsWe discuss the well-posedness and error analysis of the Biot's Poroelastic equations in this talk. To demonstrate the solvability of the poroelastic continuous issue, we first use the well-known Fredholm Alternative. In order to improve computational efficiency and address the issues raised by the discrete inf-sup condition, we present a novel and stable stabilized numerical system that is tuned for equal polynomial order. We also perform a numerical analysis to determine the stability of solutions and offer an a priori analysis of them. Lastly, we provide a few numerical illustrations. These examples offer strong proof of the usefulness and effectiveness of the suggested numerical framework.
Dienstag, 14. 05. 2024, 14:00 Uhr (WIAS-ESH)
Marwa Zainelabdeen (WIAS)
Physics-informed neural networks for convection-dominated convection-diffusion problemsIn the convection-dominated regime, solutions of convection-diffusion problems usually possess layers, which are regions where the solution has a steep gradient. It is well known that many classical numerical discretization techniques face difficulties when approximating the solution to these problems. In recent years, physics-informed neural networks (PINNs) for approximating the solution to (initial-)boundary value problems received a lot of interest. PINNs are trained to minimize several residuals (loss functionals) of the problem in collocation points. In this talk, we introduce various loss functionals for PINNs that are especially designed for convection-dominated convection-diffusion problems. They are numerically tested and compared on two benchmark problems whose solutions possess different types of layers. We observed that certain special functionals reduce the L^2 (Ω) error compared to the standard residual loss functional. Furthermore, three types of collocation points applied to hardconstrained PINNs are compared: layer-adapted, equispaced and uniformly randomly chosen points. We observe that layer-adapted points work the best for a problem with an interior layer and the worst for a problem with boundary layers.
Dienstag, 07. 05. 2024, 13:30 Uhr (WIAS-ESH)
Dilara Abdel (WIAS)
Modeling and simulation of vacancy-assisted charge transport in innovative semiconductor devicesIn response to the climate crisis, there is a need for technological innovations to reduce the escalating CO$_2$ emissions. Two promising semiconductor technologies in this regard, perovskite-based solar cells and memristive devices based on two-dimensional layered transition metal dichalcogenide (TMDC), can potentially contribute to the expansion of renewable energy sources and the development of energy-efficient computing hardware. Within perovskite and TMDC materials, ions dislocate from their ideal position in the semiconductor crystal and leave void spaces. So far, the precise influence of these vacancies and their dynamics on device performance remain underexplored.
Therefore, this talk is dedicated to comprehensively examining the impact of vacancy-assisted charge transport in innovative semiconductor devices through a theoretical approach by modeling and simulating systems of partial differential equations. We start by deriving drift-diffusion equations using thermodynamic principles. Furthermore, we formulate drift-diffusion models to describe charge transport in perovskite solar cells and TMDC memristors. We discretize the transport equations via the finite volume method and establish the existence of discrete solutions. Our study concludes with simulations conducted with an open source software tool developed in the programming language Julia. These simulations explore the influence of volume exclusion effects on charge transport in perovskite solar cells and compare our simulation results with experimental measurements found in literature for TMDC-based memristive devices.
Dienstag, 07. 05. 2024, 14:00 Uhr (WIAS-ESH)
Timo Streckenbach (WIAS)
Alternative approach to model lattice mismatch of semiconductor nanostructuresAbout using MFC (multi freedom constraints) to connect nanostructures with different lattice constants in FEM strain simulation.
Donnerstag, 02. 05. 2024, 14:00 Uhr (WIAS-ESH)
Baptiste Moreau (WIAS)
Donnerstag, 02. 05. 2024, 14:30 Uhr (WIAS-ESH)
Ondřej Pártl (WIAS)
Optimization of geothermal energy production from fracture-controlled reservoirs via 3D numerical modeling and simulationWe develop a computation framework from scratch that allows us to conduct 3D numerical simulations of groundwater flow and heat transport in hot fractured-controlled reservoirs to find optimal placements of injection and production wells that sustainably optimize geothermal energy production.
We model the reservoirs as geologically consistent randomly generated discrete fracture networks (DFN) in which the fractures are 2D manifolds with polygonal boundary embedded in a 3D porous medium. The wells are modeled as line sources and sinks. The flow and heat transport in the DFN-matrix system are modeled by solving the balance equations for mass, momentum, and energy. The fully developed computational framework combines the finite element method with semi-implicit time-stepping and algebraic flux correction. To perform the optimization, we use various gradient-free algorithms.
I will present our latest results for several geologically and physically realistic scenarios.
Dienstag, 30. 04. 2024, 13:30 Uhr (WIAS-ESH)
Patricio Farrell (WIAS)
Numerical methods for innovative semiconductor devicesThe Leibniz group NUMSEMIC develops and numerically solves nonlinear PDE models. These models are often inspired by charge transport in innovative semiconductor devices. In particular, applications include perovskite solar cells, memristors, nanowires, quantum wells, lasers as well as doping reconstruction. To translate these applications into mathematical models, we rely on nonlinear drift-diffusion, hyperelastic material models, inverse PDE problems, localized landscape theory and atomistic coupling. Our methodologies include physics-preserving finite volume methods, data-driven techniques as well as meshfree methods.
Donnerstag, 25. 04. 2024, 14:00 Uhr (WIAS-ESH)
Holger Stephan & trainees (WIAS)
Modelling of drift diffusion problems from different perspectivesJoined work with Mihaela Karcheva, Leo Markmann, Liam Johnen und Fedor Romanov
Balance equations such as drift-diffusion equations are phenomenological equations that describe the interaction of temporal changes in extensive and spatial changes in intensive physical quantities. Typical problems with such drift-diffusion models are missing oscillations and reaching the steady state after an infinitely long time. We try to overcome these problems with atypical approaches.
Dienstag, 23. 04. 2024, 13:30 Uhr (WIAS-ESH)
Zeina Amer (WIAS)
Numerical methods for coupled drift-diffusion and Helmholtz models for laser applicationsSemiconductor lasers are pivotal components in modern technologies, spanning medical procedures, manufacturing, and autonomous systems like LiDARs. Understanding their operation and developing simulation tools are paramount for advancing such technologies. In this talk, we present a mathematical PDE model for an edge-emitting laser, combining charge transport and light propagation. The charge transport will be described by a drift-diffusion model and the light propagation by the Helmholtz equation. We discuss a coupling strategy for both models and showcase initial numerical simulations.
Dienstag, 23. 04. 2024, 14:00 Uhr (WIAS-ESH)
Yiannis Hadjimichael (WIAS)
An energy-based finite-strain model for 3D heterostructured materialsThis talk presents a mathematical model that accurately describes the intrinsic strain response of 3D heterostructures arising from lattice mismatch. Combining materials with different lattice constants induces strain, leading to the bending of such heterostructures. To validate our model, we apply it to bimetallic beams and hexagonal hetero-nanowires and perform numerical simulations using finite element methods (FEM). In order to assess the fidelity of the model and the accuracy of simulations, we compare the calculated curvature with analytically derived formulations. The outcomes of our study yield valuable insights into the behavior of strained bent heterostructures. We compare the strain profiles of wurtzite and zincblende crystal structures, shedding light on their distinct characteristics. This is particularly significant as the strain has the potential to influence piezoelectricity, the electronic band structure, and the dynamics of charge carriers.
Donnerstag, 18. 04. 2024, 14:00 Uhr (WIAS-ESH)
Volker John (WIAS/FU Berlin)
Finite element methods respecting the discrete maximum principle for convection-diffusion equationsConvection-diffusion-reaction equations model the conservation of scalar quantities. From the analytic point of view, solution of these equations satisfy under certain conditions maximum principles, which represent physical bounds of the solution. That the same bounds are respected by numerical approximations of the solution is often of utmost importance in practice. The mathematical formulation of this property, which contributes to the physical consistency of a method, is called Discrete Maximum Principle (DMP). In many applications, convection dominates diffusion by several orders of magnitude. It is well known that standard discretizations typically do not satisfy the DMP in this convection-dominated regime. In fact, in this case, it turns out to be a challenging problem to construct discretizations that, on the one hand, respect the DMP and, on the other hand, compute accurate solutions. This paper presents a survey on finite element methods, with a main focus on the convection-dominated regime, that satisfy a local or a global DMP. The concepts of the underlying numerical analysis are discussed. The survey reveals that for the steady-state problem there are only a few discretizations, all of them nonlinear, that at the same time satisfy the DMP and compute reasonably accurate solutions, e.g., algebraically stabilized schemes. Moreover, most of these discretizations have been developed in recent years, showing the enormous progress that has been achieved lately. Methods based on algebraic stabilization, nonlinear and linear ones, are currently as well the only finite element methods that combine the satisfaction of the global DMP and accurate numerical results for the evolutionary equations in the convection-dominated situation.