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  • 1. Abdulle, Assyr
    et al.
    Henning, Patrick
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Numerical Analysis, NA.
    Localized orthogonal decomposition method for the wave equation with a continuum of scales2017In: Mathematics of Computation, ISSN 0025-5718, E-ISSN 1088-6842, Vol. 86, no 304, p. 549-587Article in journal (Refereed)
    Abstract [en]

    This paper is devoted to numerical approximations for the wave equation with a multiscale character. Our approach is formulated in the framework of the Localized Orthogonal Decomposition (LOD) interpreted as a numerical homogenization with an L2-projection. We derive explicit convergence rates of the method in the L∞(L2)-, W1,∞(L2)-and L∞(H1)-norms without any assumptions on higher order space regularity or scale-separation. The order of the convergence rates depends on further graded assumptions on the initial data. We also prove the convergence of the method in the framework of G-convergence without any structural assumptions on the initial data, i.e. without assuming that it is well-prepared. This rigorously justifies the method. Finally, the performance of the method is demonstrated in numerical experiments.

  • 2.
    Bastian, Peter
    et al.
    University of Heidelberg.
    Berninger, Heiko
    Dedner, Andreas
    University of Warwick.
    Engwer, Christian
    University of Münster.
    Henning, Patrick
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Numerical Analysis, NA.
    Kornhuber, Ralf
    FU Berlin.
    Kröner, Dietmar
    University of Freiburg.
    Ohlberger, Mario
    University of Münster.
    Sander, Oliver
    TU Dresden.
    Schiffler, Gerd
    Shokina, Nina
    Smetana, Kathrin
    University of Münster.
    Adaptive Modelling of Coupled Hydrological Processes with Application in Water Management2012In: Progress in Industrial Mathematics at ECMI 2010 / [ed] Michael Günther and Andreas Bartel and Markus Brunk and Sebastian Schöps and Michael Striebel, Springer Berlin/Heidelberg, 2012, p. 561-567Conference paper (Refereed)
    Abstract [en]

    This paper presents recent results of a network project aiming at the modelling and simulation of coupled surface and subsurface flows. In particular, a discontinuous Galerkin method for the shallow water equations has been developed which includes a special treatment of wetting and drying. A robust solver for saturated–unsaturated groundwater flow in homogeneous soil is at hand, which, by domain decomposition techniques, can be reused as a subdomain solver for flow in heterogeneous soil. Coupling of surface and subsurface processes is implemented based on a heterogeneous nonlinear Dirichlet–Neumann method, using the dune-grid-glue module in the numerics software DUNE.

  • 3. Engwer, C.
    et al.
    Henning, Patrick
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Numerical Analysis, NA.
    Målqvist, A.
    Peterseim, D.
    Efficient implementation of the localized orthogonal decomposition method2019In: Computer Methods in Applied Mechanics and Engineering, ISSN 0045-7825, E-ISSN 1879-2138, Vol. 350, p. 123-153Article in journal (Refereed)
    Abstract [en]

    In this paper we present algorithms for an efficient implementation of the Localized Orthogonal Decomposition method (LOD). The LOD is a multiscale method for the numerical simulation of partial differential equations with a continuum of inseparable scales. We show how the method can be implemented in a fairly standard Finite Element framework and discuss its realization for different types of problems, such as linear elliptic problems with rough coefficients and linear eigenvalue problems.

  • 4.
    Gallistl, Dietmar
    et al.
    Univ Twente, Fac Elect Engn Math & Comp Sci, POB 217, NL-7500 AE Enschede, Netherlands..
    Henning, Patrick
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Numerical Analysis, NA.
    Verfuerth, Barbara
    Westfalische Wilhelms Univ Munster, Appl Math, D-48149 Munster, Germany..
    NUMERICAL HOMOGENIZATION OF H(CURL)-PROBLEMS2018In: SIAM Journal on Numerical Analysis, ISSN 0036-1429, E-ISSN 1095-7170, Vol. 56, no 3, p. 1570-1596Article in journal (Refereed)
    Abstract [en]

    If an elliptic differential operator associated with an H (curl)- problem involves rough (rapidly varying) coefficients, then solutions to the corresponding H (curl)- problem admit typically very low regularity, which leads to arbitrarily bad convergence rates for conventional numerical schemes. The goal of this paper is to show that the missing regularity can be compensated through a corrector operator. More precisely, we consider the lowest-order Nedelec finite element space and show the existence of a linear corrector operator with four central properties: it is computable, H (curl)- stable, and quasi-local and allows for a correction of coarse finite element functions so that first-order estimates (in terms of the coarse mesh size) in the H (curl) norm are obtained provided the right-hand side belongs to H (div). With these four properties, a practical application is to construct generalized finite element spaces which can be straightforwardly used in a Galerkin method. In particular, this characterizes a homogenized solution and a first-order corrector, including corresponding quantitative error estimates without the requirement of scale separation. The constructed generalized finite element method falls into the class of localized orthogonal decomposition methods, which have not been studied for H (curl)- problems so far.

  • 5. Hellman, Fredrik
    et al.
    Henning, Patrick
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Numerical Analysis, NA.
    Malqvist, Axel
    Multiscale mixed finite elements2016In: Discrete and Continuous Dynamical Systems. Series S, ISSN 1937-1632, E-ISSN 1937-1179, Vol. 9, no 5, p. 1269-1298Article in journal (Refereed)
    Abstract [en]

    In this work, we propose a mixed finite element method for solving elliptic multiscale problems based on a localized orthogonal decomposition (LOD) of Raviart-Thomas finite element spaces. It requires to solve local problems in small patches around the elements of a coarse grid. These computations can be perfectly parallelized and are cheap to perform. Using the results of these patch problems, we construct a low dimensional multiscale mixed finite element space with very high approximation properties. This space can be used for solving the original saddle point problem in an efficient way. We prove convergence of our approach, independent of structural assumptions or scale separation. Finally, we demonstrate the applicability of our method by presenting a variety of numerical experiments, including a comparison with an MsFEM approach.

  • 6.
    Henning, Patrick
    et al.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Numerical Analysis, NA.
    Ohlberger, M.
    Verfürth, B.
    A new heterogeneous multiscale method for time-harmonic Maxwell's equations2016In: SIAM Journal on Numerical Analysis, ISSN 0036-1429, E-ISSN 1095-7170, Vol. 54, no 6, p. 3493-3522Article in journal (Refereed)
    Abstract [en]

    In this paper, we suggest a new heterogeneous multiscale method (HMM) for the time-harmonic Maxwell equations in locally periodic media. The method is constructed by using a divergence-regularization in one of the cell problems. This allows us to introduce fine-scale correctors that are not subject to a cumbersome divergence-free constraint and which can hence easily be implemented. To analyze the method, we first revisit classical homogenization theory for time-harmonic Maxwell equations and derive a new homogenization result that makes use of the divergence-regularization in the two-scale homogenized equation. We then show that the HMM is equivalent to a discretization of this equation. In particular, writing both problems in a fully coupled two-scale formulation is the crucial starting point for a corresponding numerical analysis of the method. With this approach we are able to prove rigorous a priori error estimates in the H(curl)-and the H-1-norm, and we derive reliable and efficient localized residual-based a posteriori error estimates. Numerical experiments are presented to verify the a priori convergence results.

  • 7.
    Henning, Patrick
    et al.
    Universität Münster,Germany.
    Ohlberger, Mario
    University of Münster, Germany.
    A note on homogenization of advection-diffusion problems with large expected drift2011In: Zeitschrift für Analysis und ihre Anwendungen, ISSN 0232-2064, E-ISSN 1661-4534, Vol. 30, no 3, p. 319-339Article in journal (Refereed)
  • 8.
    Henning, Patrick
    et al.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Numerical Analysis, NA.
    Ohlberger, Mario
    A-posteriori error estimate for a heterogeneous multiscale approximation of advection-diffusion problems with large expected drift2016In: Discrete and Continuous Dynamical Systems. Series S, ISSN 1937-1632, E-ISSN 1937-1179, Vol. 9, no 5, p. 1393-1420Article in journal (Refereed)
    Abstract [en]

    In this contribution we address a-posteriori error estimation in L-infinity(L-2) for a heterogeneous multiscale finite element approximation of time dependent advection-diffusion problems with rapidly oscillating coefficient functions and with a large expected drift. Based on the error estimate, we derive an algorithm for an adaptive mesh refinement. The estimate and the algorithm are validated in numerical experiments, showing applicability and good results even for heterogeneous microstructures.

  • 9.
    Henning, Patrick
    et al.
    Universität Münster, Germany.
    Ohlberger, Mario
    University of Münster, Germany.
    The heterogeneous multiscale finite element method for advection-diffusion problems with rapidly oscillating coefficients and large expected drift2010In: Networks and Heterogeneous Media, ISSN 1556-1801, E-ISSN 1556-181X, Vol. 5, no 4, p. 711-744Article in journal (Refereed)
  • 10.
    Henning, Patrick
    et al.
    University of Münster, Germany.
    Ohlberger, Mario
    University of Münster, Germany.
    The heterogeneous multiscale finite element method for elliptic homogenization problems in perforated domains2009In: Numerische Mathematik, ISSN 0029-599X, E-ISSN 0945-3245, Vol. 113, no 4, p. 601-629Article in journal (Refereed)
  • 11.
    Henning, Patrick
    et al.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Numerical Analysis, NA.
    Persson, A.
    A multiscale method for linear elasticity reducing Poisson locking2016In: Computer Methods in Applied Mechanics and Engineering, ISSN 0045-7825, E-ISSN 1879-2138, Vol. 310, p. 156-171Article in journal (Refereed)
    Abstract [en]

    We propose a generalized finite element method for linear elasticity equations with highly varying and oscillating coefficients. The method is formulated in the framework of localized orthogonal decomposition techniques introduced by Målqvist and Peterseim (2014). Assuming only L∞-coefficients we prove linear convergence in the H1-norm, also for materials with large Lamé parameter λ. The theoretical a priori error estimate is confirmed by numerical examples.

  • 12.
    Henning, Patrick
    et al.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Numerical Analysis, NA.
    Peterseim, D.
    Crank-Nicolson Galerkin approximations to nonlinear Schrödinger equations with rough potentials2017In: Mathematical Models and Methods in Applied Sciences, ISSN 0218-2025, Vol. 27, no 11, p. 2147-2184Article in journal (Refereed)
    Abstract [en]

    This paper analyzes the numerical solution of a class of nonlinear Schrödinger equations by Galerkin finite elements in space and a mass and energy conserving variant of the Crank-Nicolson method due to Sanz-Serna in time. The novel aspects of the analysis are the incorporation of rough, discontinuous potentials in the context of weak and strong disorder, the consideration of some general class of nonlinearities, and the proof of convergence with rates in L∞(L2) under moderate regularity assumptions that are compatible with discontinuous potentials. For sufficiently smooth potentials, the rates are optimal without any coupling condition between the time step size and the spatial mesh width.

  • 13.
    Henning, Patrick
    et al.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Numerical Analysis, NA.
    Wärnegård, Johan
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Numerical Analysis, NA.
    NUMERICAL COMPARISON OF MASS-CONSERVATIVE SCHEMES FOR THE GROSS-PITAEVSKII EQUATION2019In: Kinetic and Related Models, ISSN 1937-5093, E-ISSN 1937-5077, Vol. 12, no 6, p. 1247-1271Article in journal (Refereed)
    Abstract [en]

    In this paper we present a numerical comparison of various mass-conservative discretizations for the time-dependent Gross-Pitaevskii equation. We have three main objectives. First, we want to clarify how purely mass-conservative methods perform compared to methods that are additionally energy-conservative or symplectic. Second, we shall compare the accuracy of energy-conservative and symplectic methods among each other. Third, we will investigate if a linearized energy-conserving method suffers from a loss of accuracy compared to an approach which requires to solve a full nonlinear problem in each time-step. In order to obtain a representative comparison, our numerical experiments cover different physically relevant test cases, such as traveling solitons, stationary multi-solitons, Bose-Einstein condensates in an optical lattice and vortex pattern in a rapidly rotating superfluid. We shall also consider a computationally severe test case involving a pseudo Mott insulator. Our space discretization is based on finite elements throughout the paper. We will also give special attention to long time behavior and possible coupling conditions between time-step sizes and mesh sizes. The main observation of this paper is that mass conservation alone will not lead to a competitive method in complex settings. Furthermore, energy-conserving and symplectic methods are both reliable and accurate, yet, the energy-conservative schemes achieve a visibly higher accuracy in our test cases. Finally, the scheme that performs best throughout our experiments is an energy-conserving relaxation scheme with linear time-stepping proposed by C. Besse (SINUM,42(3):934-952,2004).

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