Lehrenfeld, Christoph

[["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","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"]]

[["dc.bibliographiccitation.firstpage","313"],["dc.bibliographiccitation.issue","2"],["dc.bibliographiccitation.journal","Numerische Mathematik"],["dc.bibliographiccitation.lastpage","332"],["dc.bibliographiccitation.volume","135"],["dc.contributor.author","Lehrenfeld, Christoph"],["dc.contributor.author","Reusken, Arnold"],["dc.date.accessioned","2020-03-02T16:31:02Z"],["dc.date.available","2020-03-02T16:31:02Z"],["dc.date.issued","2016"],["dc.description.abstract","In the past decade, a combination of unfitted finite elements (or XFEM) with the Nitsche method has become a popular discretization method for elliptic interface problems. This development started with the introduction and analysis of this Nitsche-XFEM technique in the paper [A. Hansbo, P. Hansbo, Comput. Methods Appl. Mech. Engrg. 191 (2002)]. In general, the resulting linear systems have very large condition numbers, which depend not only on the mesh size $, but also on how the interface intersects the mesh. This paper is concerned with the design and analysis of optimal preconditioners for such linear systems. We propose an additive subspace preconditioner which is optimal in the sense that the resulting condition number is independent of the mesh size $ and the interface position. We further show that already the simple diagonal scaling of the stifness matrix results in a condition number that is bounded by ^{-2}$, with a constant $ that does not depend on the location of the interface. Both results are proven for the two-dimensional case. Results of numerical experiments in two and three dimensions are presented, which illustrate the quality of the preconditioner."],["dc.identifier.doi","10.1007/s00211-016-0801-6"],["dc.identifier.uri","https://resolver.sub.uni-goettingen.de/purl?gro-2/63059"],["dc.language.iso","en"],["dc.relation.issn","0029-599X"],["dc.relation.issn","0945-3245"],["dc.title","Optimal preconditioners for Nitsche-XFEM discretizations of interface problems"],["dc.type","journal_article"],["dc.type.internalPublication","no"],["dspace.entity.type","Publication"]]

[["dc.bibliographiccitation.firstpage","228"],["dc.bibliographiccitation.issue","1"],["dc.bibliographiccitation.journal","SIAM Journal on Numerical Analysis"],["dc.bibliographiccitation.lastpage","255"],["dc.bibliographiccitation.volume","56"],["dc.contributor.author","Grande, Jörg"],["dc.contributor.author","Lehrenfeld, Christoph"],["dc.contributor.author","Reusken, Arnold"],["dc.date.accessioned","2020-12-10T18:37:20Z"],["dc.date.available","2020-12-10T18:37:20Z"],["dc.date.issued","2018"],["dc.identifier.doi","10.1137/16M1102203"],["dc.identifier.eissn","1095-7170"],["dc.identifier.issn","0036-1429"],["dc.identifier.uri","https://resolver.sub.uni-goettingen.de/purl?gro-2/76916"],["dc.language.iso","en"],["dc.notes.intern","DOI Import GROB-354"],["dc.relation.orgunit","Institut für Numerische und Angewandte Mathematik"],["dc.relation.workinggroup","RG Lehrenfeld (Computational PDEs)"],["dc.title","Analysis of a High-Order Trace Finite Element Method for PDEs on Level Set Surfaces"],["dc.type","journal_article"],["dc.type.internalPublication","yes"],["dspace.entity.type","Publication"]]

[["dc.bibliographiccitation.firstpage","85"],["dc.bibliographiccitation.issue","2"],["dc.bibliographiccitation.journal","Journal of Numerical Mathematics"],["dc.bibliographiccitation.lastpage","99"],["dc.bibliographiccitation.volume","27"],["dc.contributor.author","Lehrenfeld, Christoph"],["dc.contributor.author","Reusken, Arnold"],["dc.date.accessioned","2020-03-02T16:17:40Z"],["dc.date.available","2020-03-02T16:17:40Z"],["dc.date.issued","2019"],["dc.description.abstract","n 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. Recently a new unfitted finite element method was introduced which achieves a high order approximation of the geometry for domains which are implicitly described by smooth level set functions. This method is based on a parametric mapping which transforms a piecewise planar interface (or surface) reconstruction to a high order approximation. In the paper [C. Lehrenfeld and A. Reusken, IMA J. Numer. Anal. 38 (2018), No. 3, 1351–1387] an a priori error analysis of the method applied to an interface problem is presented. The analysis reveals optimal order discretization error bounds in the H1-norm. In this paper we extend this analysis and derive optimal L2-error bounds."],["dc.identifier.doi","10.1515/jnma-2017-0109"],["dc.identifier.uri","https://resolver.sub.uni-goettingen.de/purl?gro-2/63050"],["dc.language.iso","en"],["dc.relation.issn","1570-2820"],["dc.relation.issn","1569-3953"],["dc.relation.orgunit","Institut für Numerische und Angewandte Mathematik"],["dc.relation.workinggroup","RG Lehrenfeld (Computational PDEs)"],["dc.title","L2-error analysis of an isoparametric unfitted finite element method for elliptic interface problems"],["dc.type","journal_article"],["dc.type.internalPublication","yes"],["dspace.entity.type","Publication"]]