Lehrenfeld, Christoph

[["dc.bibliographiccitation.firstpage","1010"],["dc.bibliographiccitation.issue","4"],["dc.bibliographiccitation.journal","Computers & Mathematics with Applications"],["dc.bibliographiccitation.lastpage","1028"],["dc.bibliographiccitation.volume","77"],["dc.contributor.author","Schröder, Philipp W."],["dc.contributor.author","John, Volker"],["dc.contributor.author","Lederer, Philip L."],["dc.contributor.author","Lehrenfeld, Christoph"],["dc.contributor.author","Lube, Gert"],["dc.contributor.author","Schöberl, Joachim"],["dc.date.accessioned","2020-11-18T15:42:55Z"],["dc.date.available","2020-11-18T15:42:55Z"],["dc.date.issued","2018-03-19"],["dc.description.abstract","Two-dimensional Kelvin-Helmholtz instability problems are popular examples for assessing discretizations for incompressible flows at high Reynolds number. Unfortunately, the results in the literature differ considerably. This paper presents computational studies of a Kelvin-Helmholtz instability problem with high order divergence-free finite element methods. Reference results in several quantities of interest are obtained for three different Reynolds numbers up to the beginning of the final vortex pairing. A mesh-independent prediction of the final pairing is not achieved due to the sensitivity of the considered problem with respect to small perturbations. A theoretical explanation of this sensitivity to small perturbations is provided based on the theory of self-organization of 2D turbulence. Possible sources of perturbations that arise in almost any numerical simulation are discussed."],["dc.identifier.arxiv","1803.06893v4"],["dc.identifier.doi","10.1016/j.camwa.2018.10.030"],["dc.identifier.uri","https://resolver.sub.uni-goettingen.de/purl?gro-2/68807"],["dc.relation.issn","0898-1221"],["dc.relation.orgunit","Institut für Numerische und Angewandte Mathematik"],["dc.relation.workinggroup","RG Lehrenfeld (Computational PDEs)"],["dc.title","On reference solutions and the sensitivity of the 2D Kelvin-Helmholtz instability problem"],["dc.type","journal_article"],["dc.type.internalPublication","yes"],["dc.type.subtype","original_ja"],["dspace.entity.type","Publication"]]

[["dc.bibliographiccitation.firstpage","503"],["dc.bibliographiccitation.issue","2"],["dc.bibliographiccitation.journal","European Series in Applied and Industrial Mathematics. Proceedings and Surveys"],["dc.bibliographiccitation.lastpage","522"],["dc.bibliographiccitation.volume","53"],["dc.contributor.author","Lederer, Philip L."],["dc.contributor.author","Lehrenfeld, Christoph"],["dc.contributor.author","Schöberl, Joachim"],["dc.date.accessioned","2020-03-02T16:02:50Z"],["dc.date.available","2020-03-02T16:02:50Z"],["dc.date.issued","2019"],["dc.description.abstract","The present work is the second part of a pair of papers, considering Hybrid Discontinuous Galerkin methods with relaxed H(div)-conformity. The first part mainly dealt with presenting a robust analysis with respect to the mesh size h and the introduction of a reconstruction operator to restore divergence-conformity and pressure robustness (pressure independent velocity error estimates) using a modified force discretization. The aim of this part is the presentation of a high order polynomial robust analysis for the relaxed H(div)-conforming Hybrid Discontinuous Galerkin discretization of the two dimensional Stokes problem. It is based on the recently proven polynomial robust LBB-condition for BDM elements, Lederer and Schöberl (IMA J. Numer. Anal. (2017)) and is derived by a direct approach instead of using a best approximation Céa like result. We further treat the impact of the reconstruction operator on the hp analysis and present a numerical investigation considering polynomial robustness. We conclude the paper presenting an efficient operator splitting time integration scheme for the Navier–Stokes equations which is based on the methods recently presented in Lehrenfeld and Schöberl (Comp. Methods Appl. Mech. Eng. 307 (2016) 339–361) and includes the ideas of the reconstruction operator."],["dc.identifier.doi","10.1051/m2an/2018054"],["dc.identifier.uri","https://resolver.sub.uni-goettingen.de/purl?gro-2/63044"],["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","Hybrid Discontinuous Galerkin methods with relaxed H(div)-conformity for incompressible flows. Part II"],["dc.type","journal_article"],["dc.type.internalPublication","yes"],["dspace.entity.type","Publication"]]

[["dc.bibliographiccitation.firstpage","2070"],["dc.bibliographiccitation.issue","4"],["dc.bibliographiccitation.journal","SIAM Journal on Numerical Analysis"],["dc.bibliographiccitation.lastpage","2094"],["dc.bibliographiccitation.volume","56"],["dc.contributor.author","Lederer, Philip L."],["dc.contributor.author","Lehrenfeld, Christoph"],["dc.contributor.author","Schöberl, Joachim"],["dc.date.accessioned","2020-03-02T16:05:29Z"],["dc.date.available","2020-03-02T16:05:29Z"],["dc.date.issued","2018"],["dc.description.abstract","We propose a new discretization method for the Stokes equations. The method is an improved version of the method recently presented in [C. Lehrenfeld and J. Schöberl, Comp. Meth. Appl. Mech. Eng., 361 (2016)] which is based on an ({div})ehBconforming finite element space and a hybrid discontinuous Galerkin (HDG) formulation of the viscous forces. ({div})ehBconformity results in favorable properties such as pointwise divergence-free solutions and pressure robustness. However, for the approximation of the velocity with a polynomial degree $, it requires unknowns of degree $ on every facet of the mesh. In view of the superconvergence property of other HDG methods, where only unknowns of polynomial degree -1$ on the facets are required to obtain an accurate polynomial approximation of order $ (possibly after a local postprocessing), this is suboptimal. The key idea in this paper is to slightly relax the ({div})ehBconformity so that only unknowns of polynomial degree -1$ are involved for normal continuity. This allows for optimality of the method also in the sense of superconvergent HDG methods. In order not to lose the benefits of ({div})ehBconformity, we introduce a cheap reconstruction operator which restores pressure robustness and pointwise divergence-free solutions and suits well to the finite element space with relaxed ({div})ehBconformity. We present this new method, carry out a thorough ehBversion error analysis, and demonstrate the performance of the method on numerical examples. Read More: https://epubs.siam.org/doi/10.1137/17M1138078"],["dc.identifier.doi","10.1137/17M1138078"],["dc.identifier.eissn","1095-7170"],["dc.identifier.issn","0036-1429"],["dc.identifier.uri","https://resolver.sub.uni-goettingen.de/purl?gro-2/63046"],["dc.language.iso","en"],["dc.notes.intern","DOI Import GROB-354"],["dc.relation.issn","0036-1429"],["dc.relation.issn","1095-7170"],["dc.relation.orgunit","Institut für Numerische und Angewandte Mathematik"],["dc.relation.workinggroup","RG Lehrenfeld (Computational PDEs)"],["dc.title","Hybrid Discontinuous Galerkin Methods with Relaxed H(div)-Conformity for Incompressible Flows. Part I"],["dc.type","journal_article"],["dc.type.internalPublication","yes"],["dspace.entity.type","Publication"]]

[["dc.bibliographiccitation.firstpage","339"],["dc.bibliographiccitation.journal","Computer Methods in Applied Mechanics and Engineering"],["dc.bibliographiccitation.lastpage","361"],["dc.bibliographiccitation.volume","307"],["dc.contributor.author","Lehrenfeld, Christoph"],["dc.contributor.author","Schöberl, Joachim"],["dc.date.accessioned","2020-03-02T16:19:37Z"],["dc.date.available","2020-03-02T16:19:37Z"],["dc.date.issued","2015"],["dc.description.abstract","In this paper we present an efficient discretization method for the solution of the unsteady incompressible Navier-Stokes equations based on a high order (Hybrid) Discontinuous Galerkin formulation. The crucial component for the efficiency of the discretization method is the disctinction between stiff linear parts and less stiff non-linear parts with respect to their temporal and spatial treatment. Exploiting the flexibility of operator-splitting time integration schemes we combine two spatial discretizations which are tailored for two simpler sub-problems: a corresponding hyperbolic transport problem and an unsteady Stokes problem. For the hyperbolic transport problem a spatial discretization with an Upwind Discontinuous Galerkin method and an explicit treatment in the time integration scheme is rather natural and allows for an efficient implementation. The treatment of the Stokes part involves the solution of linear systems. In this case a discretization with Hybrid Discontinuous Galerkin methods is better suited. We consider such a discretization for the Stokes part with two important features: H(div)-conforming finite elements to garantuee exactly divergence-free velocity solutions and a projection operator which reduces the number of globally coupled unknowns. We present the method, discuss implementational aspects and demonstrate the performance on two and three dimensional benchmark problems."],["dc.identifier.arxiv","1508.04245v2"],["dc.identifier.doi","10.1016/j.cma.2016.04.025"],["dc.identifier.uri","https://resolver.sub.uni-goettingen.de/purl?gro-2/63052"],["dc.language.iso","en"],["dc.relation.issn","0045-7825"],["dc.title","High order exactly divergence-free Hybrid Discontinuous Galerkin Methods for unsteady incompressible flows"],["dc.type","journal_article"],["dc.type.internalPublication","no"],["dspace.entity.type","Publication"]]

[["dc.bibliographiccitation.firstpage","27"],["dc.bibliographiccitation.lastpage","56"],["dc.bibliographiccitation.volume","66"],["dc.contributor.author","Schöberl, Joachim"],["dc.contributor.author","Lehrenfeld, Christoph"],["dc.contributor.editor","Apel, T."],["dc.contributor.editor","Steinbach, O."],["dc.date.accessioned","2020-03-02T16:34:02Z"],["dc.date.available","2020-03-02T16:34:02Z"],["dc.date.issued","2013"],["dc.description.abstract","Hybrid discontinuous Galerkin methods are popular discretization methods in applications from fluid dynamics and many others. Often large scale linear systems arising from elliptic operators have to be solved. We show that standard p-version domain decomposition techniques can be applied, but we have to develop new technical tools to prove poly-logarithmic condition number estimates, in particular on tetrahedral meshes."],["dc.identifier.doi","10.1007/978-3-642-30316-6_2"],["dc.identifier.uri","https://resolver.sub.uni-goettingen.de/purl?gro-2/63060"],["dc.language.iso","en"],["dc.relation.crisseries","Lecture Notes in Applied and Computational Mechanics"],["dc.relation.doi","10.1007/978-3-642-30316-6"],["dc.relation.isbn","978-3-642-30315-9"],["dc.relation.isbn","978-3-642-30316-6"],["dc.relation.ispartof","Advanced Finite Element Methods and Applications"],["dc.relation.ispartofseries","Lecture Notes in Applied and Computational Mechanics;"],["dc.relation.issn","1613-7736"],["dc.relation.issn","1860-0816"],["dc.title","Domain Decomposition Preconditioning for High Order Hybrid Discontinuous Galerkin Methods on Tetrahedral Meshes"],["dc.type","book_chapter"],["dc.type.internalPublication","no"],["dspace.entity.type","Publication"]]