Space-Time Finite Element Methods for Parabolic Problems

Steinbach, Olaf

doi:10.1515/cmam-2015-0026

Cited by 102 publications

(104 citation statements)

References 17 publications

(28 reference statements)

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“…Recently, a space–time finite‐element method has been analyzed by Steinbach for solving the Dirichlet boundary value problem for the heat equation, as follows:

α \partial_{t} u false(x, t false) - Δ_{x} u false(x, t false) = f false(x, t false) for false(x, t false) \in Q : = normalΩ \times false(0, T false),

with given boundary and initial conditions, that is, u ( x , t )=0 for ( x , t )∈Σ:= ∂ Ω×(0, T ), and u ( x ,0)= u 0 ( x ) for x ∈Ω, respectively. Here,

normalΩ \subset R^{n}

, and therefore,

Q \subset R^{n + 1}

, n =2,3, is a bounded Lipschitz domain, and

α \in R_{+}

is the heat capacity constant.…”

Section: Adaptive Space–time Finite‐element Methodsmentioning

confidence: 99%

“…The related discrete Galerkin–Petrov variational formulation is to find

{\overline{u}}_{h} \in X_{h} \subset X

such that

a false({\overline{u}}_{h}, v_{h} false) = ⟨ f, v_{h} ⟩ - a false({\overline{u}}_{0}, v_{h} false)

is satisfied for all

v_{h} \in Y_{h} \subset Y

, where we assume

X_{h} \subset Y_{h}

. A standard stability and error analysis for the space–time finite‐element method was concluded from a discrete inf‐sup condition, as shown in the work by Steinbach . In particular, the space–time cylinder Q is decomposed into admissible and shape regular finite elements q ℓ , that is,

Q_{h} = \cup_{ℓ = 1}^{N} {\overset{q}{true}}_{ℓ}

.…”

Section: Adaptive Space–time Finite‐element Methodsmentioning

confidence: 99%

“…The finite‐element spaces are given by

X_{h} = S_{h}^{1} false(Q_{h} false) \cap X

and Y h = X h with

S_{h}^{1} false(Q_{h} false) = span {false{ϕ_{i} false}}_{i = 1}^{M}

being the span of piecewise linear and continuous basis functions ϕ i . The following energy error estimate is proved by Steinbach:

{false‖ \overset{u}{true} - {\overset{u}{true}}_{h} false‖}_{L^{2} (() 0, T; H_{0}^{1} (Ω))} \leq c h false| \overset{u}{true} |_{H^{2} false(Q false)},

where

\overset{u}{true} \in X

and

{\overset{u}{true}}_{h} \in X_{h}

denote the unique solutions of the variational problems and , respectively; c >0 is a constant independent of the mesh size h , and we assume

\overset{u}{true} \in H^{2} false(Q false)

.…”

Section: Adaptive Space–time Finite‐element Methodsmentioning

confidence: 99%

“…We aim to solve the following large sparse n × n linear system of algebraic equations:

A x = b,

arising from the space–time finite‐element variational formulation . It is clear from the work of Steinbach that the matrix A is nonsymmetric and positive definite. The linear system is solved by the preconditioned GMRES method with different AMG preconditioners.…”

Section: Amg Methodsmentioning

confidence: 99%

“…Space-time finite-element methods have received more and more interest since the pioneering work of Hughes et al 1 ; see some recent works. [2][3][4][5][6][7][8][9][10][11][12][13][14][15] A common difficulty of these methods concerns the efficient solution of the related large-scale linear system of algebraic equations, which is often harder than the solution of the linear system from more conventional time-stepping methods. Recently, in other works, 9,11,12 robust parallel geometric multigrid methods for parabolic problems using a discontinuous Galerkin space-time finite-element approach have been proposed, where a special coarsening strategy is determined by a certain precise criterion.…”

Section: Introductionmentioning

confidence: 99%

See 4 more Smart Citations

Comparison of algebraic multigrid methods for an adaptive space–time finite‐element discretization of the heat equation in 3D and 4D

Steinbach

Yang

2018

Numerical Linear Algebra App

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Summary The aim of this work is to compare algebraic multigrid (AMG) preconditioned GMRES methods for solving the nonsymmetric and positive definite linear systems of algebraic equations that arise from a space–time finite‐element discretization of the heat equation in 3D and 4D space–time domains. The finite‐element discretization is based on a Galerkin–Petrov variational formulation employing piecewise linear finite elements simultaneously in space and time. We focus on a performance comparison of conventional and modern AMG methods for such finite‐element equations, as well as robustness with respect to the mesh discretization and the heat capacity constant. We discuss different coarsening and interpolation strategies in the AMG methods for coarse‐grid selection and coarse‐grid matrix construction. Further, we compare AMG performance for the space–time finite‐element discretization on both uniform and adaptive meshes consisting of tetrahedra and pentachora in 3D and 4D, respectively. The mesh adaptivity occurring in space and time is guided by a residual‐based a posteriori error estimation.

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“…Recently, a space–time finite‐element method has been analyzed by Steinbach for solving the Dirichlet boundary value problem for the heat equation, as follows:

α \partial_{t} u false(x, t false) - Δ_{x} u false(x, t false) = f false(x, t false) for false(x, t false) \in Q : = normalΩ \times false(0, T false),

with given boundary and initial conditions, that is, u ( x , t )=0 for ( x , t )∈Σ:= ∂ Ω×(0, T ), and u ( x ,0)= u 0 ( x ) for x ∈Ω, respectively. Here,

normalΩ \subset R^{n}

, and therefore,

Q \subset R^{n + 1}

, n =2,3, is a bounded Lipschitz domain, and

α \in R_{+}

is the heat capacity constant.…”

Section: Adaptive Space–time Finite‐element Methodsmentioning

confidence: 99%

“…The related discrete Galerkin–Petrov variational formulation is to find

{\overline{u}}_{h} \in X_{h} \subset X

such that

a false({\overline{u}}_{h}, v_{h} false) = ⟨ f, v_{h} ⟩ - a false({\overline{u}}_{0}, v_{h} false)

is satisfied for all

v_{h} \in Y_{h} \subset Y

, where we assume

X_{h} \subset Y_{h}

Q_{h} = \cup_{ℓ = 1}^{N} {\overset{q}{true}}_{ℓ}

.…”

Section: Adaptive Space–time Finite‐element Methodsmentioning

confidence: 99%

“…The finite‐element spaces are given by

X_{h} = S_{h}^{1} false(Q_{h} false) \cap X

and Y h = X h with

S_{h}^{1} false(Q_{h} false) = span {false{ϕ_{i} false}}_{i = 1}^{M}

being the span of piecewise linear and continuous basis functions ϕ i . The following energy error estimate is proved by Steinbach:

{false‖ \overset{u}{true} - {\overset{u}{true}}_{h} false‖}_{L^{2} (() 0, T; H_{0}^{1} (Ω))} \leq c h false| \overset{u}{true} |_{H^{2} false(Q false)},

where

\overset{u}{true} \in X

and

{\overset{u}{true}}_{h} \in X_{h}

denote the unique solutions of the variational problems and , respectively; c >0 is a constant independent of the mesh size h , and we assume

\overset{u}{true} \in H^{2} false(Q false)

.…”

Section: Adaptive Space–time Finite‐element Methodsmentioning

confidence: 99%

“…We aim to solve the following large sparse n × n linear system of algebraic equations:

A x = b,

Section: Amg Methodsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Comparison of algebraic multigrid methods for an adaptive space–time finite‐element discretization of the heat equation in 3D and 4D

Steinbach

Yang

2018

Numerical Linear Algebra App

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A unified time discontinuous Galerkin space‐time finite element scheme for non‐Newtonian power law models

Toulopoulos

2023

Numerical Methods in Fluids

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In this work, a stabilized time Discontinuous Galerkin, Space‐Time Finite Element (tDG‐ST‐FE) scheme is presented for discretizing time‐dependent viscous shear‐thinning fluid flow models, which exhibit a usual power‐law stress strain relation. The development of the proposed numerical scheme based mainly on a unified weak space‐time formulation, where simple streamline‐upwind terms have been added in the numerical scheme, for stabilizing the discretization of the associated temporal and convective terms. The original time interval is partitioned into time subintervals, resulting in a subdivision of the space‐time cylinder into space‐time subdomains. Discontinuous Galerkin techniques are applied for the time discretization between the space‐time subdomain interfaces. A stability bound is given for the derived ST‐FE scheme. In the last part numerical examples on benchmark problems are presented for testing the efficiency of the proposed method.

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Space‐time finite element adaptive AMG for multi‐term time fractional advection diffusion equations

Yue

Liu

Shu

et al. 2019

Math Methods in App Sciences

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In this study we construct a time-space finite element (FE) scheme and furnish cost-efficient approximations for one-dimensional multi-term time fractional advection diffusion equations on a bounded domain Ω. Firstly, a fully discrete scheme is obtained by the linear FE method in both temporal and spatial directions, and many characterizations on the resulting matrix are established. Secondly, the condition number estimation is proved, an adaptive algebraic multigrid (AMG) method is further developed to lessen computational cost and analyzed in the classical framework. Finally, some numerical experiments are implemented to reach the saturation error order in the L 2 (Ω) norm sense, and present theoretical confirmations and predictable behaviors of the proposed algorithm.Another important note is that no matter which fully discrete scheme is utilized, there always exists the computational challenge in nonlocality caused by fractional differential operators [27]. 15 Quite many scholars are working to identify algorithms most appropriate to overcome the challenge and utilize computer resources. Multigrid exploits a hierarchy of grids or multiscale representations, and reduces the error of the approximation at a number of frequencies (from global smooth to local oscillation) simultaneously [28-31], which makes multigrid as an extremely superior solver or preconditioner of particular interest. Pang and Sun presented an efficient V-cycle geometric multi-20 grid (GMG) with fast Fourier transform (FFT) to solve one-dimensional space-fractional diffusion equations (SFDEs) discretized by an implicit FD scheme [32]. Bu et al. extended and analyzed the V-cycle GMG for one-dimensional MTFADEs via the FD in temporal and FE in spatial directions [26]. Zhou and Wu discussed the FE F-cycle GMG to linear stationary FADEs in Riemann-Liouvlle

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Space-Time Finite Element Methods for Parabolic Problems

Cited by 102 publications

References 17 publications

Comparison of algebraic multigrid methods for an adaptive space–time finite‐element discretization of the heat equation in 3D and 4D

Comparison of algebraic multigrid methods for an adaptive space–time finite‐element discretization of the heat equation in 3D and 4D

A unified time discontinuous Galerkin space‐time finite element scheme for non‐Newtonian power law models

Space‐time finite element adaptive AMG for multi‐term time fractional advection diffusion equations

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