Computational PDEs

Publications

Recent

recent
[HLSvW22] Fabian Heimann, Christoph Lehrenfeld, Paul Stocker, and Henry von Wahl. Unfitted Trefftz discontinuous Galerkin methods for elliptic boundary value problems. 2022. [ bib | DOI | arXiv | code | http | .pdf ]
Keywords: Numerical Analysis (math.NA), Unfitted FEM, Trefftz DG, Aggregated FEM
[BFG+22] Tobias Boege, René Fritze, Christiane Görgen, Jeroen Hanselman, Dorothea Iglezakis, Lars Kastner, Thomas Koprucki, Tabea Krause, Christoph Lehrenfeld, Silvia Polla, Marco Reidelbach, Christian Riedel, Jens Saak, Björn Schembera, Karsten Tabelow, and Marcus Weber. Research-Data Management Planning in the German Mathematical Community. 2022. [ bib | DOI | arXiv | http | .pdf ]
Keywords: History and Overview (math.HO), FOS: Mathematics, FOS: Mathematics
[HLS22] Martin Halla, Christoph Lehrenfeld, and Paul Stocker. A new T-compatibility condition and its application to the discretization of the damped time-harmonic Galbrun's equation. arXiv preprint arxiv:2209.01878, 2022. [ bib | arXiv | http | .pdf ]
Keywords: Numerical Analysis (math.NA), FOS: Mathematics, FOS: Mathematics
[LS22] Christoph Lehrenfeld and Paul Stocker. Embedded Trefftz Discontinuous Galerkin methods. arXiv preprint arXiv:2201.07041, 2022. [ bib | arXiv | code | http | .pdf ]
Keywords: Trefftz DG methods, Embedded Trefftz DG methods
[HLP22] Fabian Heimann, Christoph Lehrenfeld, and Janosch Preuß. Geometrically Higher Order Unfitted Space-Time Methods for PDEs on Moving Domains. arXiv preprint arXiv:2202.02216, 2022. [ bib | arXiv | code | http | .pdf ]
Keywords: Unfitted FEM, Space-Time FEM, Isoparametric Unfitted FEM, Higher order FEM, Petrov-Galerkin, Galerkin Collocation,
[AHLS22] Tilman Alemán, Martin Halla, Christoph Lehrenfeld, and Paul Stocker. Robust finite element discretizations for a simplified Galbrun's equation. arXiv preprint arxiv:2205.15650, 2022. [ bib | arXiv | code | http | .pdf ]
Keywords: Galbrun's equation, FEM, discontinuous Galerkin, Helmholtz decomposition

Journal publications

journal
[LL22] Yimin Lou and Christoph Lehrenfeld. Isoparametric unfitted BDF -- finite element method for PDEs on evolving domains. SIAM Journal on Numerical Analysis, 60(4):2069--2098, 2022. [ bib | DOI | code | http | .pdf ]
Keywords: Eulerian time stepping, isoparametric FEM, unfitted FEM, evolving domains,ghost penalty, stabilization, higher order FEM, BDF, projection errors
[Sto22] Paul Stocker. NGSTrefftz: Add-on to NGSolve for Trefftz methods. Journal of Open Source Software, 7(71):4135, 2022. [ bib | DOI | code | http ]
Keywords: numerical software, open source software, NGSolve, Trefftz DG, higher order FEM
[HLP21] Thorsten Hohage, Christoph Lehrenfeld, and Janosch Preuß. Learned infinite elements. SIAM J. Sci. Comput., 43(5):A3552--A3579, 2021. [ bib | DOI | code ]
Keywords: transparent boundary conditions, Dirichlet-to-Neumann map, helioseismology, learning, infinite elements, rational approximation, Helmholtz equation
[LHPvW21] Christoph Lehrenfeld, Fabian Heimann, Janosch Preuß, and Henry von Wahl. ngsxfem: Add-on to NGSolve for geometrically unfitted finite element discretizations. Journal of Open Source Software, 6(64):3237, 2021. [ bib | DOI | code | http ]
Keywords: numerical software, open source software, NGSolve, unfitted FEM, level set geometries, space-time, higher order FEM
[vWRL21] Henry von Wahl, Thomas Richter, and Christoph Lehrenfeld. An unfitted Eulerian finite element method for the time-dependent Stokes problem on moving domains. IMA Journal of Numerical Analysis, 07 2021. [ bib | DOI | arXiv | code | http | .pdf ]
[FLLS21] Guosheng Fu, Christoph Lehrenfeld, Alexander Linke, and Timo Streckenbach. Locking free and gradient robust H(div)-conforming HDG methods for linear elasticity. Journal of Scientific Computing, (39), 2021. [ bib | DOI | arXiv | http | .pdf ]
Keywords: linear elasticity, nearly incompressible, locking phenomenon, volumelocking, gradient-robustness, Discontinuous Galerkin, H(div)-conforming HDG methods
[PHL20] Janosch Preuß, Thorsten Hohage, and Christoph Lehrenfeld. Sweeping preconditioners for stratified media in the presence of reflections. Springer Nature Partial Differential Equations and Applications, 1, 2020. [ bib | DOI | arXiv | code | http | .pdf ]
Keywords: Helmholtz equation, Dirichlet-to-Neumann operator, Preconditioning, Domain decomposition, High-frequency waves, Computational seismology, Perfectly matched layers, Sweeping preconditioner
[LLS20] Philip L. Lederer, Christoph Lehrenfeld, and Joachim Schöberl. Divergence-free tangential finite element methods for incompressible flows on surfaces. 121(11):2503--2533, 2020. [ bib | DOI | arXiv | code | http | http ]
Keywords: divergence-conforming finite elements, incompressible Navier-Stokes equations, Piola transformation, surface PDEs, tangential vector field
[FKL+19] Niklas Fehn, Martin Kronbichler, Christoph Lehrenfeld, Gert Lube, and Philipp W. Schroeder. High-order DG solvers for under-resolved turbulent incompressible flows: A comparison of L2 and H(div) methods. International Journal for Numerical Methods in Fluids, 91(11):533--556, 2019. [ bib | DOI | http | .pdf ]
Keywords: discontinuous Galerkin, high-order finite elements, incompressible Navier-Stokes equations, Taylor-Green vortex, turbulent channel flow, underresolved turbulent flows
[vWRL+19] Henry von Wahl, Thomas Richter, Christoph Lehrenfeld, Jan Heiland, and Piotr Minakowski. Numerical benchmarking of fluid-rigid body interactions. Computers & Fluids, page 104290, 2019. [ bib | code | http ]
Keywords: Benchmarking, Computational fluid dynamics, Fluid–structure interaction, Finite elements, Code validation, Reference values
[LR18a] Christoph Lehrenfeld and Stephan Rave. Mass conservative reduced order modeling of a free boundary osmotic cell swelling problem. Advances in Computational Mathematics, pages 1--25, 2018. [ bib | code | http | .pdf ]
[LO19] Christoph Lehrenfeld and Maxim A. Olshanskii. An Eulerian finite element method for PDEs in time-dependent domains. ESAIM: M2AN, 53:585--614, 2019. [ bib | DOI | http | .pdf ]
[LLS19] Philip L. Lederer, Christoph Lehrenfeld, and Joachim Schöberl. Hybrid discontinuous Galerkin methods with relaxed H(div)-conformity for incompressible flows. part ii. ESAIM: M2AN, 53:503--522, 2019. [ bib | DOI | http | .pdf ]
[SJL+19] Philipp W. Schroeder, Volker John, Philip L. Lederer, Christoph Lehrenfeld, Gert Lube, and Joachim Schöberl. On reference solutions and the sensitivity of the 2D Kelvin--Helmholtz instability problem. Computers & Mathematics with Applications, 77(4):1010--1028, 2019. [ bib ]
[LOX18] Christoph Lehrenfeld, Maxim A. Olshanskii, and Xianmin Xu. A stabilized trace finite element method for partial differential equations on evolving surfaces. SIAM J. Numer. Anal., 56:1643--1672, 2018. [ bib | DOI | http | .pdf ]
[LLS18] Philip L. Lederer, Christoph Lehrenfeld, and Joachim Schöberl. Hybrid discontinuous Galerkin methods with relaxed H(div)-conformity for incompressible flows. part i. SIAM J. Numer. Anal., 56:2070--2094, 2018. [ bib | DOI | http | .pdf ]
[dPLM18] Frits de Prenter, Christoph Lehrenfeld, and André Massing. A note on the penalty parameter in Nitsche's method for unfitted boundary value problems. Computers and Mathematics with Applications, 75:4322--4336, 2018. [ bib | DOI | http | .pdf ]
[SLLL18] Philipp W. Schroeder, Alexander Linke, Christoph Lehrenfeld, and Gert Lube. Towards computable flows and robust estimates for inf-sup stable FEM applied to the time-dependent incompressible Navier-Stokes equations. SeMA Journal, 75(4):629--653, April 2018. [ bib | DOI | http ]
[FL] Guosheng Fu and Christoph Lehrenfeld. A strongly conservative hybrid DG/mixed FEM for the coupling of Stokes and Darcy flow. Journal of Scientific Computing, 77:1605--1620. [ bib | DOI | http ]
[LR18c] Christoph Lehrenfeld and Arnold Reusken. L2-estimates for a high order unfitted finite element method for elliptic interface problems. Journal of Numerical Mathematics, 00:--, 2018. first online. [ bib | DOI | http | .pdf ]
[GLR18] Jörg Grande, Christoph Lehrenfeld, and Arnold Reusken. Analysis of a high-order trace finite element method for PDEs on level set surfaces. SIAM Journal on Numerical Analysis, 56(1):228--255, 2018. [ bib | http | .pdf ]
[LR18b] Christoph Lehrenfeld and Arnold Reusken. Analysis of a high order unfitted finite element method for an elliptic interface problem. IMA J. Numer. Anal., 38:1351--1387, 2018. [ bib | DOI | http | .pdf ]
[LR17] Christoph Lehrenfeld and Arnold Reusken. Optimal preconditioners for Nitsche-XFEM discretizations of interface problems. Numerische Mathematik, 135:313--332, 2017. [ bib ]
[LS16] Christoph Lehrenfeld and Joachim Schöberl. High order exactly divergence-free hybrid discontinuous Galerkin methods for unsteady incompressible flows. Computer Methods in Applied Mechanics and Engineering, 307:339 -- 361, 2016. [ bib | DOI | http | .pdf ]
[Leh16] Christoph Lehrenfeld. High order unfitted finite element methods on level set domains using isoparametric mappings. Computer Methods in Applied Mechanics and Engineering, 300:716 -- 733, 2016. [ bib | DOI | http | .pdf ]
[DLM+16] Carlos J. Falconi D., Christoph Lehrenfeld, Holger Marschall, Christoph Meyer, R. Abiev, Dieter Bothe, Arnold Reusken, Michael Schlüter, and Martin Wörner. Numerical and experimental analysis of local flow phenomena in laminar Taylor flow in a square mini-channel. Physics of Fluids, 28(1), 2016. [ bib | DOI | http ]
[Leh15] Christoph Lehrenfeld. The Nitsche XFEM-DG space-time method and its implementation in three space dimensions. SIAM J. Sci. Comp., 37(1):A245--A270, 2015. [ bib | DOI | http | .pdf ]
[MBL+14] Holger Marschall, Stephan Boden, Christoph Lehrenfeld, Carlos J. Falconi D., Uwe Hampel, Arnold Reusken, Martin Wörner, and Dieter Bothe. Validation of interface capturing and tracking techniques with different surface tension treatments against a Taylor bubble benchmark problem. Computers & Fluids, 102:336 -- 352, 2014. [ bib | DOI | http ]
[LR13] Christoph Lehrenfeld and Arnold Reusken. Analysis of a Nitsche XFEM-DG discretization for a class of two-phase mass transport problems. SIAM J. Numer. Anal., 51:958--983, 2013. [ bib | DOI | http | .pdf ]
[LR12] Christoph Lehrenfeld and Arnold Reusken. Nitsche-XFEM with streamline diffusion stabilization for a two-phase mass transport problem. SIAM J. Sci. Comp., 34:2740--2759, 2012. [ bib | DOI | http | .pdf ]

Book chapters

bookchap
[Leh17] Christoph Lehrenfeld. A higher order isoparametric fictitious domain method for level set domains. In Stéphane P. A. Bordas, Erik Burman, Mats G. Larson, and Maxim A. Olshanskii, editors, Geometrically Unfitted Finite Element Methods and Applications, pages 65--92. Springer International Publishing, 2017. [ bib | DOI ]
[MFL+17] Holger Marschall, Carlos Falconi, Christoph Lehrenfeld, Rufat Abiev, Martin Wörner, Arnold Reusken, and Dieter Bothe. Direct numerical simulations of Taylor bubbles in a square mini-channel: Detailed shape and flow analysis with experimental validation. In Transport Processes at Fluidic Interfaces, pages 663--679. Springer, 2017. [ bib ]
[LR17] Christoph Lehrenfeld and Arnold Reusken. High order unfitted finite element methods for interface problems and PDEs on surfaces. In Transport Processes at Fluidic Interfaces, pages 33--63. Springer, 2017. [ bib | .pdf ]
[LR15] Christoph Lehrenfeld and Arnold Reusken. Finite element techniques for the numerical simulation of two-phase flows with mass transport. In Computational Methods for Complex Liquid-Fluid Interfaces, pages 353--372, 2015. [ bib | .pdf ]
[SL13] Joachim Schöberl and Christoph Lehrenfeld. Domain decomposition preconditioning for high order hybrid discontinuous Galerkin methods on tetrahedral meshes. In Thomas Apel and Olaf Steinbach, editors, Advanced Finite Element Methods and Applications, volume 66 of Lecture Notes in Applied and Computational Mechanics, pages 27--56. Springer Berlin Heidelberg, 2013. [ bib | DOI | http | .pdf ]
[KLS12] Christoph Koutschan, Christoph Lehrenfeld, and Joachim Schöberl. Computer algebra meets finite elements: An efficient implementation for Maxwells equations. In Numerical and Symbolic Scientific Computing, volume 1 of Texts and Monographs in Symbolic Computation, pages 105--121. Springer Vienna, 2012. [ bib | DOI | .pdf ]

Proceeding papers

proc
[LLS19] Christoph Lehrenfeld, Gert Lube, and Philipp W. Schroeder. A natural decomposition of viscous dissipation in DG methods for turbulent incompressible flows. PAMM, 2019. [ bib | DOI | http | .pdf ]
[HL19] Fabian Heimann and Christoph Lehrenfeld. Numerical integration on hyperrectangles in isoparametric unfitted finite elements. In European Conference on Numerical Mathematics and Advanced Applications, pages 193--202. Springer, 2019. [ bib ]
[Leh17] Christoph Lehrenfeld. Higher order unfitted finite element methods for interface problems. In Oberwolfach report, 6/2017. [ bib | .pdf ]
[LPWL16] Philip L. Lederer, Carl-Martin Pfeiler, Christoph Wintersteiger, and Christoph Lehrenfeld. Higher order unfitted FEM for Stokes interface problems. PAMM, 16(1):7--10, 2016. [ bib | DOI | http ]
[Leh16] Christoph Lehrenfeld. Removing the stabilization parameter in fitted and unfitted symmetric Nitsche formulations. In Proc. of ECCOMAS 2016, 2016. [ bib | .pdf ]
[ALM+13] Sebastian Aland, Christoph Lehrenfeld, Holger Marschall, Christoph Meyer, and Stephan Weller. Accuracy of two-phase flow simulations: The Taylor flow benchmark. Proc. Appl. Math. Mech., 13(1):595--598, December 2013. [ bib | DOI | http ]
[Leh11] Christoph Lehrenfeld. Nitsche-XFEM for a transport problem in two-phase incompressible flows. PAMM, 11(1):613--614, December 2011. [ bib | DOI | http ]

Theses

PhD theses

phdtheses
[Lou23] Yimin Lou. High-order Unfitted Discretizations for Partial Differential Equations Coupled with Geometric Flow. PhD thesis, University of Göttingen, February 2023. [ bib | http ]
[Pre21] Janosch Preuß. Learned infinite elements for helioseismology. PhD thesis, University of Göttingen, December 2021. [ bib | http ]
[Leh15] Christoph Lehrenfeld. On a Space-Time Extended Finite Element Method for the Solution of a Class of Two-Phase Mass Transport Problems. PhD thesis, RWTH Aachen, February 2015. [ bib | http | .pdf ]

Selected Master's theses

matheses
[Rau18] Hans-Georg Raumer. Shape Optimization for Interface Problems using unfitted Finite Elements. Master's thesis, NAM, University of Göttingen, February 2018. [ bib | .pdf ]
[Pre18] Janosch Preuß. Higher order unfitted isoparametric space-time FEM on moving domains. Master's thesis, NAM, University of Göttingen, February 2018. [ bib | DOI | http ]
[Jin19] Xingren Jin. Higher order stabilized time stepping in unfitted finite element method on moving domains. Master's thesis, NAM, University of Göttingen, 2019. [ bib | .pdf ]
[Hei20] Fabian Heimann. On Discontinuous- and Continuous-In-Time Unfitted Space-Time Methods for PDEs on Moving Domains. Master's thesis, NAM, University of Göttingen, October 2020. [ bib | DOI | http ]
[Ibr21] Abdul Qadir Ibrahim. Boundary layer enriched Hybrid Discontinuous Galerkin Methods for Convection dominated flow. Master's thesis, NAM, University of Göttingen, June 2021. [ bib ]
[Sch21] Anton Schmid. An HDG method for the Spalart-Allmaras model. Master's thesis, NAM, University of Göttingen, November 2021. [ bib ]
[Kem22] Michelle Kemper. Pure Eulerian unfitted FEM for Biological Fluid-Structure Interaction Problems. Master's thesis, NAM, University of Göttingen, February 2022. [ bib | DOI | http ]
[Rai22] Parajal Rai. Model Order Reduction for incompressible flows based on structure preserving discretizations. Master's thesis, NAM, University of Göttingen, February 2022. [ bib | DOI | code | http ]
[Wen22] Anna Clara Wendler. Monolithic Unfitted Space-Time FEM for an Osmotic Cell Swelling Problem. Master's thesis, NAM, University of Göttingen, February 2022. [ bib | DOI | code | http ]
[Ale22] Tilman Alemán. Robust Finite Element Discretizations for a PDE arising in Helioseismology. Master's thesis, NAM, University of Göttingen, March 2022. [ bib | DOI | code | http ]
[Kol23] Dilini Bhagya Vishwabhakthi Kolombage. Spectral Deferred Correction Methods for Spatially Discretized Flow Problems. Master's thesis, NAM, University of Göttingen, January 2023. [ bib | DOI | code | http ]
[Leh10] Christoph Lehrenfeld. Hybrid Discontinuous Galerkin Methods for Incompressible Flow Problems. Master's thesis, RWTH Aachen, May 2010. [ bib | DOI | http ]

Selected Bachelor's theses

batheses
[Kah22] Anna Kahle. The Virtual Element Method for Poisson’s equation in two-space dimensions, 2022. [ bib | DOI | http ]
[van21] Tim van Beeck. On a Discontinuous Galerkin discretization for a degenerate diffusion equation, 2021. [ bib | DOI | http ]
[Hei18] Fabian Heimann. Higher order Discontinuous Galerkin methods for the Laplace-Beltrami problem on unfitted smooth surfaces, 2018. [ bib | DOI | http ]
[Rau17] Vivian Raulin. Model order reduction for linear PDE problems with the reduced basis method, 2017. [ bib | .pdf ]