Research WG Applications of PDEs - Prof. Dr. Christian Engwer

  • Overview and focus

    Many of our applications origin from porous media or biological systems, which exhibit very different kinds of complexity. Complexity can origin from a complicated geometric shapes, which poses a challenge for the numerical solution of PDEs in the complex shaped domain. The other kind of complexity is complexity of the system itself, due to complex couplings between different physical, biological & chemical processes.

    Complex Geometries

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     When solving of PDEs on time dependent domains or domains with a complex shape, classic Finite Element Methods pose many problems regarding the construction of the mesh. During the last decade a range of different methods have been developed to decouple the construction of a finite element mesh, i.e. the finite element discretization, from the geometrical details of the domain.

    One approach, our group is working on, is the Unfitted Discontinuous Galerkin method. It offers the possibility to compute simulations with a fine structures on a relatively coarse mesh and was used for the solution of elliptic, parabolic and hyperbolic problems. Using the UDG approach it is easily possible to run simulations directly on image data, e.g. micro-CT images, or to combine it with level-set or phase-field methods to handle moving interfaces.

    Multi-Physics Problems

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    In the course of multi-physics applications the efficient coupling of different PDEs on different sub-domains is getting more important. We are working on different aspects of domain decomposition methods and their implementation, either for parallelization and preconditioning, or for the coupling in a multi-physics setting. The latter also includes heterogenous coupling of sub-domains of different dimension.

    Efficient PDE Software

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    Dune
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    We set high value on the development of efficient FEM software. Reusability and interoperability of and with existing software are very important. In this course we are actively participating in the development of the C++ FEM framework DUNE.

    Programming with C++ and using generic programming techniques, allows us to use fine grained interfaces and still employ optimization techniques, like inlining and loop-unrolling. This is the basis for sustainable and efficient software development.

    High Performance Computing

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    The speed of a single processor stopped growing in the last years, instead modern chips include many cores to increase the performance. At the same time the architecture of high performance computers like the BlueGene is changing, they include acceleration processors which leaves us with a heterogeneous hardware system. Modern scientific software must cope with these changing requirements. As it is too much a burden to expect scientist to rewrite their code for each new hardware, the software design and the numerical algorithms must be adopted in a way that allows us to port our software with as small work as possible, while still retaining a reasonable performance boost.

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    Latest Publications

    • , , and . . “Model order reduction of an ultraweak and optimally stable variational formulation for parametrized reactive transport problems.” SIAM Journal on Scientific Computing, № 46 (5): A3205–A3229. doi: 10.1137/23M1613402.
    • , , and . . “Construction of local reduced spaces for Friedrichs' systems via randomized training.” contribution to the Central-European Conference on Scientific Computing, ALGORITMY, Podbanské

    • , , , , , , , , , , , , and . “Brainstorm-DUNEuro: An integrated and user-friendly Finite Element Method for modeling electromagnetic brain activity.” NeuroImage, № 267 119851. doi: 10.1016/j.neuroimage.2022.119851.
    • , , , , , , , , , , , and . “CutFEM forward modeling for EEG source analysis.” Frontiers in Human Neuroscience, № 17 1216758. doi: 10.3389/fnhum.2023.1216758.
    • , , and . . “An optimally stable approximation of reactive transport using discrete test and infinite trial spaces.” in Finite Volumes for Complex Applications X—Volume 2, Hyperbolic and Related Problems, Springer Proceedings in Mathematics & Statistics, edited by Emmanuel Franck, Jürgen Fuhrmann, Michel-Dansac Victor and Laurent Navoret. Heidelberg: Springer. doi: 10.1007/978-3-031-40860-1_30.

    • , , , , , , , , , , and . . “The DUNE Framework: Basic Concepts and Recent Developments.” Computers & Mathematics with Applications, № 81: 75–112. doi: 10.1016/j.camwa.2020.06.007.
    • , , , and . . “Monotonicity considerations for stabilized DG cut cell schemes for the unsteady advection equation.” contribution to the ENUMATH2019, Egmond aan Zee, The Netherlands
    • . . “Strategies for the vectorized Block Conjugate Gradients method.” contribution to the ENUMATH2019, Egmond aan Zee, The Netherlands
    • , , , , , , , , and . . “DUNEuro- A software toolbox for forward modeling in bioelectromagnetism.” PloS one, № 2021 doi: 10.1371/journal.pone.0252431.
    • . . “Hardware-Oriented Krylov Methods for High-Performance Computing.” Dissertation thesis, WWU Münster. doi: 10.48550/arXiv.2104.02494.

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    Supervised Doctoral Studies

    Wenske, MichaelData-aware methods for the simulation of glioblastoma multiforme
    An unfitted discontinuous Galerkin scheme for a phase-field approximation of pressurized fractures
    Towards Automatic and Reliable Localized Model Order Reduction. Local Training, a Posteriori Error Estimation and Online Enrichment.
    Piastra, Maria CarlaNew Finite Element Methods for MEG and combined EEG/MEG Forward Problem
    Fitted and unfitted finite element methods for solving the EEG forward problem
    Emken, NatalieA coupled bulk-surface reaction-diffusion-advection model for cell polarization
    Vorwerk, JohannesNew Finite Element Methods to Solve the EEG/MEG Forward Problem
    Vorwerk, JohannesNew Finite Element Methods to solve the EEG/MEG Forward Problem
    DTI data based multiscale modelling and simulation of glioma growth