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

    • 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.’ arXiv e-prints 2019: arXiv:1909.13672. [Submitted]

    • Dreier Nils-Arne, Engwer Christian, Hartmann Dirk. . ‘Galerkin local maximum entropy method.’ International Journal for Numerical Methods in Fluids 0. doi: 10.1002/fld.4513.
    • Engwer Christian, Altenbernd Mirco, Dreier Nils-Arne, Dominik Göddeke. . ‘A high-level C++ approach to manage local errors, asynchrony and faults in an MPI application.’ Contributed to the 26th Euromicro International Conference on Parallel, Distributed and Network-based Processing, Cambridge, Vereinigtes Königreich. doi: doi.org/10.1109/PDP2018.2018.00117.

    • Vorwerk J, Engwer C, Pursiainen S, Wolters CH. . ‘A Mixed Finite Element Method to Solve the EEG Forward Problem.’ IEEE Transactions on Medical Imaging 36, No. 4: 930-941. doi: 10.1109/TMI.2016.2624634.
    • Engwer C, Nüßing A. . ‘Geometric Reconstruction of Implicitly Defined Surfaces and Domains with Topological Guarantees.’ ACM Transactions on Mathematical Software 44, No. 2. doi: 10.1145/3104989.
    • Engwer C, Vorwerk J, Ludewig J, Wolters CH. . ‘A Discontinuous Galerkin Method for the EEG Forward Problem.’ SIAM Journal on Scientific Computing 39, No. 1: B138-B164. doi: 10.1137/15M1048392.
    • Buhr A, Engwer C, Ohlberger M, Rave S. . ‘ArbiLoMod: Local Solution Spaces by Random Training in Electrodynamics.’ In Model Reduction of Parametrized Systems, edited by Benner P., Ohlberger M., Patera A., Rozza G., Urban K., 137-148. Cham: Springer International Publishing. doi: 10.1007/978-3-319-58786-8_9.
    • Buhr A, Engwer C, Ohlberger M, Rave S. . ‘ArbiLoMod, a Simulation Technique Designed for Arbitrary Local Modifications.’ SIAM J. Sci. Comput. 39, No. 4: A1435–A1465. doi: 10.1137/15M1054213.
    • Grebhahn A, Engwer C, Bolten M, Apel S. . ‘Variability of Stencil Computations for Porous Media.’ Concurrency and Computation: Practice and Experience Special Issue on Advanced Stencil-Code Engineering. [In Press]
    • Engwer C, Falgout R, Yang UM. . ‘A Framework of Stencil Computations for PDE based Applications with Examples from DUNE and hypre.’ Concurrency and Computation: Practice and Experience Special Issue on Advanced Stencil-Code Engineering. [In Press]

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    Dissertations

    Piastra, Maria CarlaNew Finite Element Methods for MEG and combined EEG/MEG Forward Problem
    Nüßing, AndreasFitted 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
    Knappitsch, MarkusDTI data based multiscale modelling and simulation of glioma growth