arXiv subjects

Fabian Heimann

Publications and source records attributed to Fabian Heimann.

7 recordsLinked to original sources

arXiv2026

Local error estimates for a finite element method combining linear and nonlinear stabilization for the linear hyperbolic transport equation

In this paper, we investigate the combination of a linear continuous interior penalty type and a non-linear artificial diffusion stabilisation applied to the transport problem, based on continuous Galerkin finite elements in space. This method was recently introduced and analysed for globally smooth solutions in [Burman 2023, SIAM J. Sci. Comput., 45, 1, A96-A122]. We provide a rigorous proof of a localisation principle in terms of weighted stability and a priori error bound results, which follow the widely known $\mathcal{O}(h^{k+1/2})$ scaling in the $L^2(Ω; t=T)$ norm, where $k$ denotes the polynomial order of the finite element space and $h$ the mesh size. The analysis is semi-discrete in space and assumes sufficient local regularity of the continuous solution on the smooth part of the domain, where the continuous interior penalty stabilisation is active, whilst artificial diffusion operates on the remaining rough parts of the domain. Thereby, the analysis demonstrates that typical numerical errors in the rough part stay localised relative to the convection velocity and do not negatively affect the smooth parts of the solution, if the stabilisation combination is set up accordingly.

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arXiv2025

Higher Order Unfitted Space-Time Methods for Transport Problems

In this article, we present an Unfitted Space-Time Finite Element method for the scalar transport equation posed on moving domains. We consider the case of the domain boundary being transported by the same velocity field as the scalar concentration inside the physical domain. A standard continuous Galerkin Finite element space is considered on a fixed background mesh, as well as tensor product Space-Time elements, which can be discontinuous along time slice boundaries. For the computational geometry, we opt for a spatially second-order accurate approximation variant in the mathematical analysis. In particular, we establish stability in a problem-specific norm and prove a priori error bounds of high order. Numerical examples illustrate these theoretical findings.

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arXiv2025

Discretization Error Analysis of a High Order Unfitted Space-Time Method for moving domain problems

We present a numerical analysis of a higher order unfitted space-time Finite Element method applied to a convection-diffusion model problem posed on a moving bulk domain. The method uses isoparametric space-time mappings for the geometry approximation of level set domains and has been presented and investigated computationally in [Heimann, Lehrenfeld, Preuß, SIAM J. Sci. Comp. 45(2), 2023, B139 - B165]. Recently, in [Heimann, Lehrenfeld, IMA J. Numer. Anal., 2025] error bounds for the geometry approximation have been proven. In this paper we prove stability and accuracy including the influence of the geometry approximation.

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arXiv2024

Geometry Error Analysis of a parametric mapping for Higher Order Unfitted Space-Time Methods

In [Heimann, Lehrenfeld, Preuß, SIAM J. Sci. Comp. 45(2), 2023, B139 - B165] new geometrically unfitted space-time Finite Element methods for partial differential equations posed on moving domains of higher-order accuracy in space and time have been introduced. For geometrically higher-order accuracy a parametric mapping on a background space-time tensor-product mesh has been used. In this paper, we concentrate on the geometrical accuracy of the approximation and derive rigorous bounds for the distance between the realized and an ideal mapping in different norms and derive results for the space-time regularity of the parametric mapping. These results are important and lay the ground for the error analysis of corresponding unfitted space-time finite element methods.

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arXiv2024

A Higher Order Unfitted Space-Time Finite Element Method for Coupled Surface-Bulk problems

We present a higher order space-time unfitted finite element method for convection-diffusion problems on coupled (surface and bulk) domains. In that way, we combine a method suggested by Heimann, Lehrenfeld, Preuß (SIAM J. Sci. Comput. 45(2), 2023, B139 - B165) for the bulk case with a method suggested by Sass, Reusken (Comput. Math. Appl. 146(15), 2023, 253-270) for the surface case. The geometry is allowed to change with time, and the higher order discrete approximation of this geometry is ensured by a time-dependent isoparametric mapping. The space-time discretisation approach allows for straightforward handling of arbitrary high orders. In that way, we also generalise results of Hansbo, Larson, Zahedi (Comput. Methods Appl. Mech. Engrg. 307, 2016, 96-116) to higher orders. The convergence of the proposed higher order discretisations is confirmed numerically.

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arXiv2023

Unfitted Trefftz discontinuous Galerkin methods for elliptic boundary value problems

We propose a new geometrically unfitted finite element method based on discontinuous Trefftz ansatz spaces. Trefftz methods allow for a reduction in the number of degrees of freedom in discontinuous Galerkin methods, thereby, the costs for solving arising linear systems significantly. This work shows that they are also an excellent way to reduce the number of degrees of freedom in an unfitted setting. We present a unified analysis of a class of geometrically unfitted discontinuous Galerkin methods with different stabilisation mechanisms to deal with small cuts between the geometry and the mesh. We cover stability and derive a-priori error bounds, including errors arising from geometry approximation for the class of discretisations for a model Poisson problem in a unified manner. The analysis covers Trefftz and full polynomial ansatz spaces, alike. Numerical examples validate the theoretical findings and demonstrate the potential of the approach.

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arXiv2022

Geometrically Higher Order Unfitted Space-Time Methods for PDEs on Moving Domains

In this paper, we propose new geometrically unfitted space-time Finite Element methods for partial differential equations posed on moving domains of higher order accuracy in space and time. As a model problem, the convection-diffusion problem on a moving domain is studied. For geometrically higher order accuracy, we apply a parametric mapping on a background space-time tensor-product mesh. Concerning discretisation in time, we consider discontinuous Galerkin, as well as related continuous (Petrov-)Galerkin and Galerkin collocation methods. For stabilisation with respect to bad cut configurations and as an extension mechanism that is required for the latter two schemes, a ghost penalty stabilisation is employed. The article puts an emphasis on the techniques that allow to achieve a robust but higher order geometry handling for smooth domains. We investigate the computational properties of the respective methods in a series of numerical experiments. These include studies in different dimensions for different polynomial degrees in space and time, validating the higher order accuracy in both variables.

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