Foundations of space-time finite element methods: Polytopes, interpolation, and integration

Frontin, Cory V.; Walters, Gage; Witherden, Freddie D.; Lee, Carl W.; Williams, David M.; Darmofal, David L.

doi:10.1016/j.apnum.2021.03.019

“…[13][14][15] Note that the solution of transient three-dimensional problems with space-time finite elements requires four-dimensional meshes. Recent advances in generation, 9,16,17 adaptation, 18 and numerical handling 19,20 of four-dimensional meshes mark the state-of-the-art in this active research field.…”

Section: Motivationmentioning

confidence: 99%

Time‐continuous and time‐discontinuous space‐time finite elements for advection‐diffusion problems

Danwitz

¹

,

Voulis

²

,

Hosters

³

et al. 2023

Numerical Meth Engineering

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We construct four variants of space‐time finite element discretizations based on linear tensor‐product and simplex‐type finite elements. The resulting discretizations are continuous in space, and continuous or discontinuous in time. In a first test run, all four methods are applied to a linear scalar advection‐diffusion model problem. Then, the convergence properties of the time‐discontinuous space‐time finite element discretizations are studied in numerical experiments. Advection velocity and diffusion coefficient are varied, such that the parabolic case of pure diffusion (heat equation), as well as, the hyperbolic case of pure advection (transport equation) are included in the study. For each model parameter set, the L2$$ {L}_2 $$ error at the final time is computed for spatial and temporal element lengths ranging over several orders of magnitude to allow for an individual evaluation of the methods' spatial, temporal, and space‐time accuracy. In the parabolic case, particular attention is paid to the influence of time‐dependent boundary conditions. Key findings include a spatial accuracy of second order and a temporal accuracy between second and third order. The temporal accuracy tends toward third order depending on how advection‐dominated the test case is, on the choice of the specific discretization method, and on the time‐(in)dependence and treatment of the boundary conditions. Additionally, the potential of time‐continuous simplex space‐time finite elements for heat flux computations is demonstrated with a piston ring pack test case and a subtractive manufacturing test case.

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“…For application to hyperbolic problems, see e.g. [26,31,47], advectiondiffusion problems [56,86], the compressible Euler equations [92,94], wave equations [32,57,93], the Schrödinger equation [52], the Navier-Stokes equations [55,56,73] and fractional PDEs [63].…”

Section: Further Readingmentioning

confidence: 99%

“…These results show the potential of the Dune code even for 4D problems. Recently, 4D problems have been taken into consideration [32], but to the best of our knowledge this is the first 4D DG-SEM implementation available publicly.…”

Section: Advection-diffusionmentioning

confidence: 99%

Theoretical and Practical Aspects of Space-Time DG-SEM Implementations

Versbach¹,

Linders²,

Klöfkorn³

et al. 2023

The SMAI Journal of Computational Mathematics

3

0

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We discuss two approaches for the formulation and implementation of space-time discontinuous Galerkin spectral element methods (DG-SEM). In one, time is treated as an additional coordinate direction and a Galerkin procedure is applied to the entire problem. In the other, the method of lines is used with DG-SEM in space and the fully implicit Runge-Kutta method Lobatto IIIC in time. The two approaches are mathematically equivalent in the sense that they lead to the same discrete solution. However, in practice they differ in several important respects, including the terminology used to describe them, the structure of the resulting software, and the interaction with nonlinear solvers. Challenges and merits of the two approaches are discussed with the goal of providing the practitioner with sufficient consideration to choose which path to follow. Additionally, implementations of the two methods are provided as a starting point for further development. Numerical experiments validate the theoretical accuracy of these codes and demonstrate their utility, even for 4D problems.

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“…The implementation of space-time methods with a focus on the challenge of 4D problems has been studied in [30], and the generation of different 4D space-time meshes have been presented in [6,14].…”

Section: Further Readingmentioning

confidence: 99%

“…These results show the potential of the Dune code even for 4D problems. Recently, 4D problems have been taken into consideration [30], but to the best of our knowledge this is the first 4D DG-SEM implementation available publicly.…”

Section: Euler Equationsmentioning

confidence: 99%

Theoretical and Practical Aspects of Space-Time DG-SEM Implementations

Versbach¹,

Linders²,

Klöfkorn³

et al. 2022

Preprint

0

View full text Add to dashboard Cite

We discuss two approaches for the formulation and implementation of space-time discontinuous Galerkin spectral element methods (DG-SEM). In one, time is treated as an additional coordinate direction and a Galerkin procedure is applied to the entire problem. In the other, the method of lines is used with DG-SEM in space and the fully implicit Runge-Kutta method Lobatto IIIC in time. The two approaches are mathematically equivalent in the sense that they lead to the same discrete solution. However, in practice they differ in several important respects, including the terminology used to describe them, the structure of the resulting software, and the interaction with nonlinear solvers. Challenges and merits of the two approaches are discussed with the goal of providing the practitioner with sufficient consideration to choose which path to follow. Additionally, implementations of the two methods are provided as a starting point for further development. Numerical experiments validate the theoretical accuracy of these codes and demonstrate their utility, even for 4D problems.

show abstract

Foundations of space-time finite element methods: Polytopes, interpolation, and integration

Cited by 10 publications

References 47 publications

Time‐continuous and time‐discontinuous space‐time finite elements for advection‐diffusion problems

Time‐continuous and time‐discontinuous space‐time finite elements for advection‐diffusion problems

Theoretical and Practical Aspects of Space-Time DG-SEM Implementations

Theoretical and Practical Aspects of Space-Time DG-SEM Implementations

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