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High-order space-time finite element schemes for acoustic and viscodynamic wave equations with temporal decoupling.

Banks HT, Birch MJ, Brewin MP, Greenwald SE, Hu S, Kenz ZR, Kruse C, Maischak M, Shaw S, Whiteman JR - Int J Numer Methods Eng (2014)

Bottom Line: If the spatial finite element space is of dimension D and polynomials of degree r are used in time, the block system has dimension (r + 1)D and is usually regarded as being too large when r > 1.Werder et al. found that the space-time coupling matrices are diagonalizable over [Formula: see text] for r ⩽100, and this means that the time-coupled computations within a time step can actually be decoupled.Copyright © 2014 The Authors.

View Article: PubMed Central - PubMed

Affiliation: Center for Research in Scientific Computation, North Carolina State University Raleigh, NC 27695-8212, USA.

ABSTRACT

We revisit a method originally introduced by Werder et al. (in Comput. Methods Appl. Mech. Engrg., 190:6685-6708, 2001) for temporally discontinuous Galerkin FEMs applied to a parabolic partial differential equation. In that approach, block systems arise because of the coupling of the spatial systems through inner products of the temporal basis functions. If the spatial finite element space is of dimension D and polynomials of degree r are used in time, the block system has dimension (r + 1)D and is usually regarded as being too large when r > 1. Werder et al. found that the space-time coupling matrices are diagonalizable over [Formula: see text] for r ⩽100, and this means that the time-coupled computations within a time step can actually be decoupled. By using either continuous Galerkin or spectral element methods in space, we apply this DG-in-time methodology, for the first time, to second-order wave equations including elastodynamics with and without Kelvin-Voigt and Maxwell-Zener viscoelasticity. An example set of numerical results is given to demonstrate the favourable effect on error and computational work of the moderately high-order (up to degree 7) temporal and spatio-temporal approximations, and we also touch on an application of this method to an ambitious problem related to the diagnosis of coronary artery disease. Copyright © 2014 The Authors. International Journal for Numerical Methods in Engineering published by John Wiley & Sons Ltd.

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Results for the Example 2 version of 59 showing (left) the energy error, 11, and (right) the corresponding work-error dependence for the Galerkin method.
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fig01: Results for the Example 2 version of 59 showing (left) the energy error, 11, and (right) the corresponding work-error dependence for the Galerkin method.

Mentions: This is exactly as mentioned previously for Example 1 but with E1 = 30 kPa ⋅s. The results for the computed energy error bound given by 11 are shown in Figure 1 for the Galerkin error The SEM gave a similar picture.


High-order space-time finite element schemes for acoustic and viscodynamic wave equations with temporal decoupling.

Banks HT, Birch MJ, Brewin MP, Greenwald SE, Hu S, Kenz ZR, Kruse C, Maischak M, Shaw S, Whiteman JR - Int J Numer Methods Eng (2014)

Results for the Example 2 version of 59 showing (left) the energy error, 11, and (right) the corresponding work-error dependence for the Galerkin method.
© Copyright Policy - open-access
Related In: Results  -  Collection

License
Show All Figures
getmorefigures.php?uid=PMC4376204&req=5

fig01: Results for the Example 2 version of 59 showing (left) the energy error, 11, and (right) the corresponding work-error dependence for the Galerkin method.
Mentions: This is exactly as mentioned previously for Example 1 but with E1 = 30 kPa ⋅s. The results for the computed energy error bound given by 11 are shown in Figure 1 for the Galerkin error The SEM gave a similar picture.

Bottom Line: If the spatial finite element space is of dimension D and polynomials of degree r are used in time, the block system has dimension (r + 1)D and is usually regarded as being too large when r > 1.Werder et al. found that the space-time coupling matrices are diagonalizable over [Formula: see text] for r ⩽100, and this means that the time-coupled computations within a time step can actually be decoupled.Copyright © 2014 The Authors.

View Article: PubMed Central - PubMed

Affiliation: Center for Research in Scientific Computation, North Carolina State University Raleigh, NC 27695-8212, USA.

ABSTRACT

We revisit a method originally introduced by Werder et al. (in Comput. Methods Appl. Mech. Engrg., 190:6685-6708, 2001) for temporally discontinuous Galerkin FEMs applied to a parabolic partial differential equation. In that approach, block systems arise because of the coupling of the spatial systems through inner products of the temporal basis functions. If the spatial finite element space is of dimension D and polynomials of degree r are used in time, the block system has dimension (r + 1)D and is usually regarded as being too large when r > 1. Werder et al. found that the space-time coupling matrices are diagonalizable over [Formula: see text] for r ⩽100, and this means that the time-coupled computations within a time step can actually be decoupled. By using either continuous Galerkin or spectral element methods in space, we apply this DG-in-time methodology, for the first time, to second-order wave equations including elastodynamics with and without Kelvin-Voigt and Maxwell-Zener viscoelasticity. An example set of numerical results is given to demonstrate the favourable effect on error and computational work of the moderately high-order (up to degree 7) temporal and spatio-temporal approximations, and we also touch on an application of this method to an ambitious problem related to the diagnosis of coronary artery disease. Copyright © 2014 The Authors. International Journal for Numerical Methods in Engineering published by John Wiley & Sons Ltd.

No MeSH data available.


Related in: MedlinePlus