Space-time hexahedral finite element methods for parabolic evolution problems

Langer, Ulrich; Schafelner, Andreas

doi:10.48550/arxiv.2103.13835

Cited by 1 publication

(1 citation statement)

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“…Note that this work is also extended to hexahedral space-time discretizations. 30 Moreover, Langer and Zank propose and investigate new efficient direct solvers for time-continuous tensor-product discretizations of the parabolic initial boundary value problem. 31 The influence of linear constraints, for example, time-dependent Dirichlet boundary conditions, on discontinuous Galerkin time discretization methods for parabolic problems is treated by Voulis and Reusken.…”

Section: Literature Reviewmentioning

confidence: 99%

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

Danwitz

Voulis

Hosters

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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Section: Literature Reviewmentioning

confidence: 99%