Private Homepage | https://www.uni-muenster.de/AMM/engwer/team/engwer.shtml |

Research Interests | Numerical methods for partial differential equations Scoientific Computing High-Performance Computing Design and development of numerical software Cut-cell methods Numerical methods for surface PDEs and geometric PDEs Biomedical Applications |

Project membershipMathematics Münster | C: Models and ApproximationsC1: Evolution and asymptotics C2: Multi-scale phenomena and macroscopic structures C4: Geometry-based modelling, approximation, and reduction |

Current Publications | • Medani, Takfarinas; Garcia-Prieto, Juan; Tadel, Francois; Antonakakis, Marios; Erdbrügger, Tim; Höltershinken, Malte; Mead, Wayne; Schrader, Sophie; Joshi, Anand; Engwer, Christian; Wolters, Carsten H.; Mosher, John C.; Leahy, Richard M. Brainstorm-DUNEuro: An integrated and user-friendly Finite Element Method for modeling electromagnetic brain activity. NeuroImage Vol. 267, 2023 online• Renelt, Lukas; Ohlberger, Mario; Engwer, Christian An optimally stable approximation of reactive transport using discrete test and infinite trial spaces. , 2023 online• Renelt, Lukas; Ohlberger, Mario; Engwer, Christian An optimally stable approximation of reactive transport using discrete test and infinite trial spaces. , 2023 online• Bastian P, Blatt M, Dedner A, Dreier N, Engwer C, Fritze R, Gräser C, Kempf D, Klöfkorn R, Ohlberger M, Sander O The DUNE Framework: Basic Concepts and Recent Developments. Computers & Mathematics with Applications Vol. 81, 2021, pp 75-112 online• Streitbürger Florian, Engwer Christian, May Sandra, Nüßing Andreas Monotonicity considerations for stabilized DG cut cell schemes for the unsteady advection equation. , 2021 online• Dreier Nils-Arne, Engwer Christian Strategies for the vectorized Block Conjugate Gradients method. , 2021 online• Schrader S, Westhoff A, Piastra MC, Miinalainen T, Pursiainen S,Vorwerk J, Brinck H, Wolters CH, Engwer C DUNEuro- A software toolbox for forward modeling in bioelectromagnetism. PloS one Vol. 2021, 2021 online• Bastian P, Blatt M, Dedner A, Dreier N, Engwer C, Fritze R, Gräser C, Kempf D, Klöfkorn R, Ohlberger M, Sander O The DUNE Framework: Basic Concepts and Recent Developments. Computers & Mathematics with Applications Vol. 81, 2021, pp 75-112 online• Bastian P, Altenbernd M, Dreier N, Engwer C, Fahlke J, Fritze R, Geveler M, Göddeke D, Iliev O, Ippisch O, Mohring J, Müthing S, Ohlberger M, Ribbrock D, Shegunov N, Turek S Exa-Dune -- Flexible PDE Solvers, Numerical Methods and Applications. Software for Exascale Computing - SPPEXA 2016-2019LNCSE, 2020, pp 225-269 online |

Current Projects | • Interdisziplinäres Lehrprogramm zu maschinellem Lernen und künstlicher Intelligenz The aim of the project is to establish and test a graduated university-wide teaching programme on machine learning (ML) and artificial intelligence (AI). AI is taught as an interdisciplinary cross-sectional topic that has diverse application possibilities in basic research as well as in economy and society, but consequently also raises social, ethical and ecological challenges. The modular teaching program is designed to enable students to build up their AI knowledge, apply it independently and transfer it directly to various application areas. The courses take place in a broad interdisciplinary context, i.e., students from different departments take the courses together and work together on projects. online• Personalised diagnosis and treatment for refractory focal paediatric and adult epilepsy Epilepsy is among the most common neurological diseases, affecting between 0.5% and 1% of the general population. Therefore, new diagnosis and treatment methods have a high impact on society. Epilepsy is also among the most frequently diagnosed neurological paediatric disorders, with long-term implications for the quality of life of those affected. Only in two-thirds of cases, seizures can be adequately controlled with anticonvulsant drug treatment. For the remaining drug-refractory patients with focal epilepsy (up to about 2 Mill. in Europe), epilepsy surgery is currently the most effective treatment. However, only 15-20% of those patients are eligible for epilepsy surgery. That is either because the epileptogenic zone in the brain cannot be localized with sufficient accuracy with standard diagnostic means, or because the epileptogenic zone overlaps with eloquent cortical areas, so that it cannot be surgically removed without considerable neurological deficit. PerEpi aims to bring together a group of experts at the European level to improve this situation in two ways, both of which use concepts of non-invasive personalised medicine: The first one focuses on a new individualised multimodal approach to set a new milestone in localization accuracy of the epileptogenic zone in order to offer the most appropriate personalised therapy. The second one focuses on a new individually optimized transcranial electric brain stimulation technique as a new treatment option to reduce seizure frequency and severity. This is particularly attractive for those focal refractory patients where surgery is not an option because of an overlap with eloquent cortical areas. A dedicated ethics work package will ensure that the research in the consortium is designed and conducted following the highest ethical standards. In addition, this work package will study the translational pathways of the new approaches to foster clinical integration that is ethically and socially responsible. online• HyperCut – Stabilized DG schemes for hyperbolic conservation laws on cut cell meshes The goal of this project is to develop new tools for solving time-dependent, first order hyperbolic conservation laws, in particular the compressible Euler equations, on complex shaped domains.In practical applications, mesh generation is a major issue. When dealing with complicated geometries, the construction of corresponding body-fitted meshes is a very involved and time-consuming process.In this proposal, we will consider a different approach: In the last two decades so called cut cell methods have gained a lot of interest, as they reduce the burden of the meshing process. The idea is to simply cut the geometry out of a Cartesian background mesh. Theresulting cut cells can have various shapes and are not bounded from below in size. Compared to body-fitted meshes, this approach is fully automatic and much cheaper. However, standard explicit schemes are typically not stable when the time step is chosen with respect to the background mesh and does not reflect the size of small cut cells. Thisis referred to as the small cell problem.In the setting of standard meshes, both Finite Volume (FV) and Discontinuous Galerkin (DG) methods have been used successfully for solving non-linear hyperbolic conservation laws. For FV schemes, there already exist several approaches for extending thesemethods to cut cell meshes and overcoming the small cell problem while keeping the explicit time stepping. For DG schemes, this is not the case.The goal of this proposal is to develop stable DG schemes for solving time-dependent hyperbolic conservation laws, in particular the compressible Euler equations, on cut cell meshes using explicit time stepping.We particularly aim at a method that(1) solves the small cell problem and permits explicit time stepping,(2) preserves mass conservation,(3) is high-order along the cut cell boundary, where many important quantities are evaluated,(4) satisfies theoretical properties such as monotonicity and TVDM stability for model problems,(5) works for non-linear hyperbolic conservation laws, in particular the compressible Euler equations,(6) is robust in the presence of shocks or discontinuities,(7) and sufficiently simple to be implemented in higher dimensions.We base the spatial discretization on a DG approach to enable high accuracy. We plan to develop new stabilization terms to overcome the small cell problem for this setup. The starting point for this proposal is our recent publication for stabilizing a DG discretizationfor linear advection using piecewise linear polynomials. We will extend these results in different directions, namely to non-linear problems, including the compressible Euler equations, and to higher order, in particular to piecewise quadratic polynomials.We will implement these methods using the software framework DUNE and publish our code as open-source. online• BrainStorm: Highly Extensible Software for Advanced Electrophysiology and MEG/EEG Imaging Software grant for integrating DUNEuro (http://www.duneuro.org) into Brainstorm (http://neuroimage.usc.edu/brainstorm) online• EXC 2044 - C1: Evolution and asymptotics In this unit, we will use generalisations of optimal transport metrics to develop gradient flow descriptions of (cross)-diffusion-reaction systems, rigorously analyse their pattern forming properties, and develop corresponding efficient numerical schemes. Related transport-type- and hyperbolic systems will be compared with respect to their pattern-forming behaviour, especially when mass is conserved. Bifurcations and the effects of noise perturbations will be explored.Moreover, we aim to understand defect structures, their stability and their interactions. Examples are the evolution of fractures in brittle materials and of vortices in fluids. Our analysis will explore the underlying geometry of defect dynamics such as gradient descents or Hamiltonian structures. Also, we will further develop continuum mechanics and asymptotic descriptions for multiple bodies which deform, divide, move, and dynamically attach to each other in order to better describe the bio-mechanics of growing and dividing soft tissues. Finally, we are interested in the asymptotic analysis of various random structures as the size or the dimension of the structure goes to infinity. More specifically, we shall consider random polytopes and random trees.For random polytopes we would like to compute the expected number of faces in all dimensions, the expected (intrinsic) volume, and absorption probabilities, as well as higher moments and limit distributions for these quantities. online • EXC 2044 - C2: Multi-scale phenomena and macroscopic structures In multi-scale problems, geometry and dynamics on the micro-scale influence structures on coarser scales. In this research unit we will investigate and analyse such structural interdependence based on topological, geometrical or dynamical properties of the underlying processes.We are interested in transport-dominated processes, such as in the problem of how efficient a fluid can be stirred to enhance mixing, and in the related analytical questions. A major concern will be the role of molecular diffusion and its interplay with the stirring process. High Péclet number flow in porous media with reaction at the surface of the porous material will be studied. Here, the flowinduces pore-scale fluctuations that lead to macroscopic enhanced diffusion and reaction kinetics. We also aim at understanding advection-dominated homogenisation problems in random regimes. We aim at classifying micro-scale geometry or topology with respect to the macroscopic behaviour of processes considered therein. Examples are meta material modelling and the analysis of processes in biological material. Motivated by network formation and fracture mechanics in random media, we will analyse the effective behaviour of curve and free-discontinuity energies with stochastic inhomogeneity. Furthermore, we are interested in adaptive algorithms that can balance the various design parameters arising in multi-scale methods. The analysis of such algorithms will be the key towards an optimal distribution of computational resources for multi-scale problems. Finally, we will study multi-scale energy landscapes and analyse asymptotic behaviour of hierarchical patterns occurring in variational models for transportation networks and related optimal transport problems. In particular, we will treat questions of self-similarity, cost distribution, and locality of the fine-scale pattern. We will establish new multilevel stochastic approximation algorithms with the aim of numerical optimisation in high dimensions. online • EXC 2044 - C4: Geometry-based modelling, approximation, and reduction In mathematical modelling and its application to the sciences, the notion of geometry enters in multiple related but different flavours: the geometry of the underlying space (in which e.g. data may be given), the geometry of patterns (as observed in experiments or solutions of corresponding mathematical models), or the geometry of domains (on which PDEs and their approximations act). We will develop analytical and numerical tools to understand, utilise and control geometry, also touching upon dynamically changing geometries and structural connections between different mathematical concepts, such as PDE solution manifolds, analysis of pattern formation, and geometry. online |

E-Mail | christian.engwer@uni-muenster.de |

Phone | +49 251 83-35067 |

FAX | +49 251 83-32729 |

Room | 120.020 |

Secretary | Sekretariat Wernke Frau Silvia Wernke Telefon +49 251 83-35052 Fax +49 251 83-32729 Zimmer 120.001 |

Address | Prof. Dr. Christian Engwer Angewandte Mathematik Münster: Institut für Analysis und Numerik Fachbereich Mathematik und Informatik der Universität Münster Orléans-Ring 10 48149 Münster |

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