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

Christoph Lehrenfeld; Fabian Heimann; Janosch Preu{\ss}

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

Fabian Heimann, Christoph Lehrenfeld, Janosch Preu{ss}

Numerical Analysis · Submitted 4 Feb 2022

abstract compiled · 2026-09-23 · no language-model review

The authors' main claim 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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arXiv ID2202.02216 (version 2)

Pith IDY2JALFI6, Pith's permanent record of this paper and its review

CitationChristoph Lehrenfeld, Fabian Heimann, Janosch Preu{\ss}. "Geometrically Higher Order Unfitted Space-Time Methods for PDEs on Moving Domains." arXiv:2202.02216 (2022). https://pith.science/paper/Y2JALFI6

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What the abstract itself states, as probabilities: claims first 0.04 · states limitation 0.21 · human participants 0.01 · controlled comparison 0.04 · randomized assignment 0.01 · states quantitative result 0.06 · compares with prior results 0.10 · mentions code or data availability 0.02.

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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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What the classifier reports for each sentence. The number is the probability it gave the label.
#	sentence	role	the authors' own work	states the main conclusion	strength
s0	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.	contribution 1.00	0.92	0.24	asserted 0.99
s1	As a model problem, the convection-diffusion problem on a moving domain is studied.	method 0.44	0.65	0.03	asserted 0.98
s2	For geometrically higher order accuracy, we apply a parametric mapping on a background space-time tensor-product mesh.	method 0.98	0.89	0.05	asserted 1.00
s3	Concerning discretisation in time, we consider discontinuous Galerkin, as well as related continuous (Petrov-)Galerkin and Galerkin collocation methods.	method 0.97	0.78	0.03	asserted 0.98
s4	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.	method 0.98	0.83	0.05	asserted 1.00
s5	The article puts an emphasis on the techniques that allow to achieve a robust but higher order geometry handling for smooth domains.	contribution 0.90	0.43	0.18	asserted 0.87
s6	We investigate the computational properties of the respective methods in a series of numerical experiments.	method 0.60	0.94	0.04	asserted 0.96
s7	These include studies in different dimensions for different polynomial degrees in space and time, validating the higher order accuracy in both variables.	result 0.47	0.94	0.19	asserted 0.98

What kind of paper the classifier reports this is: method or tool 1.00 · evidence from simulation or computation 0.99.

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Original abstract

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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2 arXiv papers cite this in 11 sentences

Each citing sentence is taken from the citing paper's LaTeX source and typed by Pith's classifier. A label appears at probability 0.8 or more (0.95 for support), a contest at 0.7 or more; the quoted words are the citing authors'. How each label was checked · JSON

Papers citing it · 7 sentences without a confident label

Uses its method or tool 1 paper

We explain the space-time mesh deformation on an arbitrary time slab $Q_n$, $n=1,…,N$, also used in [ref] .

An Accurate and Robust Eulerian Finite Element Method for Partial Differential Equations on Evolving Surfaces · Space-time mesh deformation · 0.92

Uses its data or values 1 paper

As in [ref] , we take $μ= 0.1$.

Space-time unfitted finite element methods for time-dependent problems on moving domains · An example with topology change · 0.99

Compares with it 1 paper

In contrast to the results reported in [ref] , we do not see any spurious diffusion starting before the time of first contact even though there is only a single full cell between the two disks at the time slab right before contact (see Figure [ref]).

Space-time unfitted finite element methods for time-dependent problems on moving domains · An example with topology change · 0.94

What it cites · 6 sentences without a confident label

Uses the method or tool of 1 paper

This does not need to be a continuous discrete function, so that we recommend to apply the previously discussed projection only element-wise and skip the final averaging step, called $P_h$ in [ref] .

Isoparametric unfitted BDF -- Finite element method for PDEs on evolving domains · Discontinuous-in-time deformations and mesh transfer operations · 0.90

Cites as background 1 paper

In characteristic-based methods also known as semi-Lagrangian methods such as those discussed in [ref] instead of the partial derivative $∂/∂t$ the material derivative $d/dt = ∂/∂t + w·∇$ is approximated.

A fourth-order unfitted characteristic finite element method for solving the advection-diffusion equation on time-varying domains · Approaches in the literature · 0.91

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