Lehrenfeld, Christoph

[["dc.bibliographiccitation.firstpage","105"],["dc.bibliographiccitation.lastpage","121"],["dc.contributor.author","Koutschan, Christoph"],["dc.contributor.author","Lehrenfeld, Christoph"],["dc.contributor.author","Schoeberl, Joachim"],["dc.contributor.editor","Langer, Ulrich"],["dc.date.accessioned","2021-03-04T07:48:22Z"],["dc.date.available","2021-03-04T07:48:22Z"],["dc.date.issued","2011-04-21"],["dc.description.abstract","We consider the numerical discretization of the time-domain Maxwell's equations with an energy-conserving discontinuous Galerkin finite element formulation. This particular formulation allows for higher order approximations of the electric and magnetic field. Special emphasis is placed on an efficient implementation which is achieved by taking advantage of recurrence properties and the tensor-product structure of the chosen shape functions. These recurrences have been derived symbolically with computer algebra methods reminiscent of the holonomic systems approach."],["dc.identifier.arxiv","1104.4208v2"],["dc.identifier.doi","10.1007/978-3-7091-0794-2_6"],["dc.identifier.uri","https://resolver.sub.uni-goettingen.de/purl?gro-2/79889"],["dc.relation.isbn","978-3-7091-0793-5"],["dc.relation.isbn","978-3-7091-0794-2"],["dc.relation.ispartof","Numerical and symbolic scientific computing: progress and prospects"],["dc.title","Computer Algebra meets Finite Elements: an Efficient Implementation for Maxwell's Equations"],["dc.type","book_chapter"],["dc.type.internalPublication","unknown"],["dspace.entity.type","Publication"]]

[["dc.bibliographiccitation.firstpage","A3552"],["dc.bibliographiccitation.issue","5"],["dc.bibliographiccitation.journal","SIAM Journal on Scientific Computing"],["dc.bibliographiccitation.lastpage","A3579"],["dc.bibliographiccitation.volume","43"],["dc.contributor.author","Hohage, Thorsten"],["dc.contributor.author","Lehrenfeld, Christoph"],["dc.contributor.author","Preuß, Janosch"],["dc.date.accessioned","2021-07-20T10:21:15Z"],["dc.date.available","2021-07-20T10:21:15Z"],["dc.date.issued","2020-10-29"],["dc.description.abstract","We study the numerical solution of scalar time-harmonic wave equations on unbounded domains which can be split into a bounded interior domain of primary interest and an exterior domain with separable geometry. To compute the solution in the interior domain, approximations to the Dirichlet-to-Neumann (DtN) map of the exterior domain have to be imposed as transparent boundary conditions on the artificial coupling boundary. Although the DtN map can be computed by separation of variables, it is a nonlocal operator with dense matrix representations, and hence computationally inefficient. Therefore, approximations of DtN maps by sparse matrices, usually involving additional degrees of freedom, have been studied intensively in the literature using a variety of approaches including different types of infinite elements, local non-reflecting boundary conditions, and perfectly matched layers. The entries of these sparse matrices are derived analytically, e.g. from transformations or asymptotic expansions of solutions to the differential equation in the exterior domain. In contrast, in this paper we propose to `learn' the matrix entries from the DtN map in its separated form by solving an optimization problem as a preprocessing step. Theoretical considerations suggest that the approximation quality of learned infinite elements improves exponentially with increasing number of infinite element degrees of freedom, which is confirmed in numerical experiments. These numerical studies also show that learned infinite elements outperform state-of-the-art methods for the Helmholtz equation. At the same time, learned infinite elements are much more flexible than traditional methods as they, e.g., work similarly well for exterior domains involving strong reflections, for example, for the atmosphere of the Sun, which is strongly inhomogeneous and exhibits reflections at the corona."],["dc.identifier.doi","10.1137/20M1381757"],["dc.identifier.uri","https://resolver.sub.uni-goettingen.de/purl?gro-2/88272"],["dc.language.iso","en"],["dc.relation","SFB 1456 | Cluster C | C04: Correlations of solar oscillations: modeling and inversions"],["dc.relation","SFB 1456 | Cluster C: Data with Information in Their Dependency Structure"],["dc.relation","SFB 1456: Mathematik des Experiments: Die Herausforderung indirekter Messungen in den Naturwissenschaften"],["dc.relation.orgunit","Institut für Numerische und Angewandte Mathematik"],["dc.relation.workinggroup","RG Lehrenfeld (Computational PDEs)"],["dc.rights","CC BY 4.0"],["dc.title","Learned infinite elements"],["dc.type","journal_article"],["dc.type.internalPublication","yes"],["dc.type.subtype","original_ja"],["dspace.entity.type","Publication"]]

[["dc.bibliographiccitation.firstpage","585"],["dc.bibliographiccitation.issue","2"],["dc.bibliographiccitation.journal","European Series in Applied and Industrial Mathematics. Proceedings and Surveys"],["dc.bibliographiccitation.lastpage","614"],["dc.bibliographiccitation.volume","53"],["dc.contributor.author","Lehrenfeld, Christoph"],["dc.contributor.author","Olshanskii, Maxim"],["dc.date.accessioned","2020-03-02T16:01:40Z"],["dc.date.available","2020-03-02T16:01:40Z"],["dc.date.issued","2019"],["dc.description.abstract","The paper introduces a new finite element numerical method for the solution of partial differential equations on evolving domains. The approach uses a completely Eulerian description of the domain motion. The physical domain is embedded in a triangulated computational domain and can overlap the time-independent background mesh in an arbitrary way. The numerical method is based on finite difference discretizations of time derivatives and a standard geometrically unfitted finite element method with an additional stabilization term in the spatial domain. The performance and analysis of the method rely on the fundamental extension result in Sobolev spaces for functions defined on bounded domains. This paper includes a complete stability and error analysis, which accounts for discretization errors resulting from finite difference and finite element approximations as well as for geometric errors coming from a possible approximate recovery of the physical domain. Several numerical examples illustrate the theory and demonstrate the practical efficiency of the method."],["dc.identifier.doi","10.1051/m2an/2018068"],["dc.identifier.uri","https://resolver.sub.uni-goettingen.de/purl?gro-2/63043"],["dc.language.iso","en"],["dc.relation.issn","0764-583X"],["dc.relation.issn","1290-3841"],["dc.relation.orgunit","Institut für Numerische und Angewandte Mathematik"],["dc.relation.workinggroup","RG Lehrenfeld (Computational PDEs)"],["dc.title","An Eulerian finite element method for PDEs in time-dependent domains"],["dc.type","journal_article"],["dc.type.internalPublication","yes"],["dspace.entity.type","Publication"]]

[["dc.bibliographiccitation.firstpage","958"],["dc.bibliographiccitation.issue","2"],["dc.bibliographiccitation.journal","SIAM Journal on Numerical Analysis"],["dc.bibliographiccitation.lastpage","983"],["dc.bibliographiccitation.volume","51"],["dc.contributor.author","Lehrenfeld, Christoph"],["dc.contributor.author","Reusken, Arnold"],["dc.date.accessioned","2021-03-05T08:59:02Z"],["dc.date.available","2021-03-05T08:59:02Z"],["dc.date.issued","2013"],["dc.identifier.doi","10.1137/120875260"],["dc.identifier.uri","https://resolver.sub.uni-goettingen.de/purl?gro-2/80333"],["dc.language.iso","en"],["dc.notes.intern","DOI Import GROB-393"],["dc.relation.eissn","1095-7170"],["dc.relation.issn","0036-1429"],["dc.title","Analysis of a Nitsche XFEM-DG Discretization for a Class of Two-Phase Mass Transport Problems"],["dc.type","journal_article"],["dc.type.internalPublication","unknown"],["dspace.entity.type","Publication"]]

[["dc.bibliographiccitation.firstpage","1351"],["dc.bibliographiccitation.issue","3"],["dc.bibliographiccitation.journal","IMA Journal of Numerical Analysis"],["dc.bibliographiccitation.lastpage","1387"],["dc.bibliographiccitation.volume","38"],["dc.contributor.author","Lehrenfeld, Christoph"],["dc.contributor.author","Reusken, Arnold"],["dc.date.accessioned","2020-03-02T16:18:28Z"],["dc.date.available","2020-03-02T16:18:28Z"],["dc.date.issued","2018"],["dc.description.abstract","In the context of unfitted finite element discretizations, the realization of high-order methods is challenging due to the fact that the geometry approximation has to be sufficiently accurate. We consider a new unfitted finite element method that achieves a high-order approximation of the geometry for domains that are implicitly described by smooth-level set functions. The method is based on a parametric mapping, which transforms a piecewise planar interface reconstruction to a high-order approximation. Both components, the piecewise planar interface reconstruction and the parametric mapping, are easy to implement. In this article, we present an a priori error analysis of the method applied to an interface problem. The analysis reveals optimal order error bounds for the geometry approximation and for the finite element approximation, for arbitrary high-order discretization. The theoretical results are confirmed in numerical experiments."],["dc.identifier.doi","10.1093/imanum/drx041"],["dc.identifier.uri","https://resolver.sub.uni-goettingen.de/purl?gro-2/63051"],["dc.language.iso","en"],["dc.relation.issn","0272-4979"],["dc.relation.issn","1464-3642"],["dc.relation.orgunit","Institut für Numerische und Angewandte Mathematik"],["dc.relation.workinggroup","RG Lehrenfeld (Computational PDEs)"],["dc.title","Analysis of a high-order unfitted finite element method for elliptic interface problems"],["dc.type","journal_article"],["dc.type.internalPublication","yes"],["dspace.entity.type","Publication"]]

[["dc.bibliographiccitation.firstpage","533"],["dc.bibliographiccitation.issue","11"],["dc.bibliographiccitation.journal","International Journal for Numerical Methods in Fluids"],["dc.bibliographiccitation.lastpage","556"],["dc.bibliographiccitation.volume","91"],["dc.contributor.author","Fehn, Niklas"],["dc.contributor.author","Kronbichler, Martin"],["dc.contributor.author","Lehrenfeld, Christoph"],["dc.contributor.author","Lube, Gert"],["dc.contributor.author","Schröder, Philipp W."],["dc.date.accessioned","2020-03-02T16:00:09Z"],["dc.date.available","2020-03-02T16:00:09Z"],["dc.date.issued","2019"],["dc.description.abstract","The accurate numerical simulation of turbulent incompressible flows is a challenging topic in computational fluid dynamics. For discretisation methods to be robust in the under-resolved regime, mass conservation as well as energy stability are key ingredients to obtain robust and accurate discretisations. Recently, two approaches have been proposed in the context of high-order discontinuous Galerkin (DG) discretisations that address these aspects differently. On the one hand, standard ^2ehBbased DG discretisations enforce mass conservation and energy stability weakly by the use of additional stabilisation terms. On the other hand, pointwise divergence-free (\\operatorname{div})ehBconforming approaches ensure exact mass conservation and energy stability by the use of tailored finite element function spaces. The present work raises the question whether and to which extent these two approaches are equivalent when applied to under-resolved turbulent flows. This comparative study highlights similarities and differences of these two approaches. The numerical results emphasise that both discretisation strategies are promising for under-resolved simulations of turbulent flows due to their inherent dissipation mechanisms."],["dc.identifier.arxiv","1905.00142v1"],["dc.identifier.doi","10.1002/fld.4763"],["dc.identifier.uri","https://resolver.sub.uni-goettingen.de/purl?gro-2/63042"],["dc.language.iso","en"],["dc.notes.intern","DeepGreen Import"],["dc.relation.issn","0271-2091"],["dc.relation.issn","1097-0363"],["dc.relation.orgunit","Institut für Numerische und Angewandte Mathematik"],["dc.relation.workinggroup","RG Lehrenfeld (Computational PDEs)"],["dc.title","High-order DG solvers for under-resolved turbulent incompressible flows: A comparison of $L^2$ and $H(div)$ methods"],["dc.type","journal_article"],["dc.type.internalPublication","yes"],["dspace.entity.type","Publication"]]

[["dc.bibliographiccitation.artnumber","012109"],["dc.bibliographiccitation.issue","1"],["dc.bibliographiccitation.journal","Physics of Fluids"],["dc.bibliographiccitation.volume","28"],["dc.contributor.author","Falconi, C. J."],["dc.contributor.author","Lehrenfeld, C."],["dc.contributor.author","Marschall, H."],["dc.contributor.author","Meyer, C."],["dc.contributor.author","Abiev, R."],["dc.contributor.author","Bothe, D."],["dc.contributor.author","Reusken, A."],["dc.contributor.author","Schlüter, M."],["dc.contributor.author","Wörner, M."],["dc.date.accessioned","2020-03-02T16:22:05Z"],["dc.date.available","2020-03-02T16:22:05Z"],["dc.date.issued","2016"],["dc.description.abstract","The vertically upward Taylor flow in a small square channel (side length 2 mm) is one of the guiding measures within the priority program “Transport Processes at Fluidic Interfaces” (SPP 1506) of the German Research Foundation (DFG). This paper presents the results of coordinated experiments and three-dimensional numerical simulations (with three different academic computer codes) for typical local flow parameters (bubble shape, thickness of the liquid film, and velocity profiles) in different cutting planes (lateral and diagonal) for a specific co-current Taylor flow. For most quantities, the differences between the three simulation results and also between the numerical and experimental results are below a few percent. The experimental and computational results consistently show interesting three-dimensional flow effects in the rear part of the liquid film. There, a local back flow of liquid occurs in the fixed frame of reference which leads to a temporary reversal of the direction of the wall shear stress during the passage of a Taylor bubble. Notably, the axial positions of the region with local backflow and those of the minimum vertical velocity differ in the lateral and the diagonal liquid films. By a thorough analysis of the fully resolved simulation results, this previously unknown phenomenon is explained in detail and, moreover, approximate criteria for its occurrence in practical applications are given. It is the different magnitude of the velocity in the lateral film and in the corner region which leads to azimuthal pressure differences in the lateral and diagonal liquid films and causes a slight deviation of the bubble from the rotational symmetry. This deviation is opposite in the front and rear parts of the bubble and has the mentioned significant effects on the local flow field in the rear part of the liquid film."],["dc.identifier.doi","10.1063/1.4939498"],["dc.identifier.uri","https://resolver.sub.uni-goettingen.de/purl?gro-2/63054"],["dc.language.iso","en"],["dc.relation.issn","1070-6631"],["dc.relation.issn","1089-7666"],["dc.title","Numerical and experimental analysis of local flow phenomena in laminar Taylor flow in a square mini-channel"],["dc.type","journal_article"],["dc.type.internalPublication","no"],["dspace.entity.type","Publication"]]

[["dc.bibliographiccitation.firstpage","629"],["dc.bibliographiccitation.issue","4"],["dc.bibliographiccitation.journal","SeMA Journal"],["dc.bibliographiccitation.lastpage","653"],["dc.bibliographiccitation.volume","75"],["dc.contributor.author","Schröder, Philipp W."],["dc.contributor.author","Lehrenfeld, Christoph"],["dc.contributor.author","Linke, Alexander"],["dc.contributor.author","Lube, Gert"],["dc.date.accessioned","2020-03-02T16:11:38Z"],["dc.date.available","2020-03-02T16:11:38Z"],["dc.date.issued","2018"],["dc.description.abstract","Inf-sup stable FEM applied to time-dependent incompressible Navier–Stokes flows are considered. The focus lies on robust estimates for the kinetic and dissipation energies in a twofold sense. Firstly, pressure–robustness ensures the fulfilment of a fundamental invariance principle and velocity error estimates are not corrupted by the pressure approximability. Secondly, Re-semi-robustness means that constants appearing on the right-hand side of kinetic and dissipation energy error estimates (including Gronwall constants) do not explicitly depend on the Reynolds number. Such estimates rely on the essential regularity assumption ∇u∈L1(0,T;L∞(Ω)) which is discussed in detail. In the sense of best practice, we review and establish pressure- and Re-semi-robust estimates for pointwise divergence-free H1-conforming FEM (like Scott–Vogelius pairs or certain isogeometric based FEM) and pointwise divergence-free H(div)-conforming discontinuous Galerkin FEM. For convection-dominated problems, the latter naturally includes an upwind stabilisation for the velocity which is not gradient-based."],["dc.identifier.doi","10.1007/s40324-018-0157-1"],["dc.identifier.uri","https://resolver.sub.uni-goettingen.de/purl?gro-2/63048"],["dc.language.iso","en"],["dc.relation.issn","2254-3902"],["dc.relation.issn","2281-7875"],["dc.relation.orgunit","Institut für Numerische und Angewandte Mathematik"],["dc.relation.workinggroup","RG Lehrenfeld (Computational PDEs)"],["dc.title","Towards computable flows and robust estimates for inf-sup stable FEM applied to the time-dependent incompressible Navier–Stokes equations"],["dc.type","journal_article"],["dc.type.internalPublication","yes"],["dspace.entity.type","Publication"]]

[["dc.bibliographiccitation.firstpage","A2740"],["dc.bibliographiccitation.issue","5"],["dc.bibliographiccitation.journal","SIAM Journal on Scientific Computing"],["dc.bibliographiccitation.lastpage","A2759"],["dc.bibliographiccitation.volume","34"],["dc.contributor.author","Lehrenfeld, Christoph"],["dc.contributor.author","Reusken, Arnold"],["dc.date.accessioned","2020-03-02T16:40:27Z"],["dc.date.available","2020-03-02T16:40:27Z"],["dc.date.issued","2012"],["dc.description.abstract","We consider an unsteady convection diffusion equation which models the transport of a dissolved species in two-phase incompressible flow problems. The so-called Henry interface condition leads to a jump condition for the concentration at the interface between the two phases. In [A. Hansbo and P. Hansbo, Comput. Methods Appl. Mech. Engrg., 191 (2002), pp. 5537--5552], for the purely elliptic stationary case, an extended finite element method (XFEM) is combined with a Nitsche-type method, and optimal error bounds are derived. These results were extended to the unsteady case in [A. Reusken and T. Nguyen, J. Fourier Anal. Appl., 15 (2009), pp. 663--683]. In the latter paper convection terms are also considered but assumed to be small. In many two-phase flow applications, however, convection is the dominant transport mechanism. Hence there is a need for a stable numerical method for the case of a convection dominated transport equation. In this paper we address this topic and study the streamline diffusion stabilization for the Nitsche-XFEM. The method is presented, and results of numerical experiments are given that indicate that this kind of stabilization is satisfactory for this problem class. Furthermore, a theoretical error analysis of the stabilized Nitsche-XFEM is presented that results in optimal a priori discretization error bounds."],["dc.identifier.doi","10.1137/110855235"],["dc.identifier.uri","https://resolver.sub.uni-goettingen.de/purl?gro-2/63063"],["dc.language.iso","en"],["dc.relation.issn","1064-8275"],["dc.relation.issn","1095-7197"],["dc.title","Nitsche-XFEM with Streamline Diffusion Stabilization for a Two-Phase Mass Transport Problem"],["dc.type","journal_article"],["dc.type.internalPublication","no"],["dspace.entity.type","Publication"]]

[["dc.bibliographiccitation.firstpage","336"],["dc.bibliographiccitation.journal","Computers & Fluids"],["dc.bibliographiccitation.lastpage","352"],["dc.bibliographiccitation.volume","102"],["dc.contributor.author","Marschall, Holger"],["dc.contributor.author","Boden, Stephan"],["dc.contributor.author","Lehrenfeld, Christoph"],["dc.contributor.author","Falconi D., Carlos J."],["dc.contributor.author","Hampel, Uwe"],["dc.contributor.author","Reusken, Arnold"],["dc.contributor.author","Wörner, Martin"],["dc.contributor.author","Bothe, Dieter"],["dc.date.accessioned","2021-03-05T08:58:05Z"],["dc.date.available","2021-03-05T08:58:05Z"],["dc.date.issued","2014"],["dc.identifier.doi","10.1016/j.compfluid.2014.06.030"],["dc.identifier.uri","https://resolver.sub.uni-goettingen.de/purl?gro-2/80002"],["dc.language.iso","en"],["dc.notes.intern","DOI Import GROB-393"],["dc.relation.issn","0045-7930"],["dc.title","Validation of Interface Capturing and Tracking techniques with different surface tension treatments against a Taylor bubble benchmark problem"],["dc.type","journal_article"],["dc.type.internalPublication","unknown"],["dspace.entity.type","Publication"]]