Limits...
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.

No MeSH data available.


Related in: MedlinePlus

Plots of the L2(Ω) error in u(T) against N (equal in space and time) for the wave equation problem in Example 5 (Galerkin-in-space).
© Copyright Policy - open-access
Related In: Results  -  Collection

License
getmorefigures.php?uid=PMC4376204&req=5

fig03: Plots of the L2(Ω) error in u(T) against N (equal in space and time) for the wave equation problem in Example 5 (Galerkin-in-space).

Mentions: Here, Dirichlet boundary data are assumed on the whole of ∂Ω and, г, the boundary and the initial data are chosen so that the exact solution is u = cos(2πt) sin(40πx) cos(30πy). The results are plotted on the left of the top row in Figures 3, 4 and 5 for the Galerkin-in-space scheme for errors in u(T), ∇ u(T) and w(T), with the estimated convergence rates tabulated on the right (Notice that a zero value of L2 error for w is reported for the CN scheme when N = 2 — this is no more than a numerical anomaly resulting from exact boundary data and a very coarse mesh.) These are based on the computed errors for N (first column) and for 2N (the results for N = 4 use those for N = 2, the row for which is not shown). The middle row of the figure shows the result of computing with one degree higher polynomials in time than in space, whereas the bottom row shows the result of using one degree higher in space than in time. The CN results are of course unaffected in the middle row. Analogous results for the spectral element scheme in the case where equal degrees are used have also been obtained and are contained in 6.


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)

Plots of the L2(Ω) error in u(T) against N (equal in space and time) for the wave equation problem in Example 5 (Galerkin-in-space).
© Copyright Policy - open-access
Related In: Results  -  Collection

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

fig03: Plots of the L2(Ω) error in u(T) against N (equal in space and time) for the wave equation problem in Example 5 (Galerkin-in-space).
Mentions: Here, Dirichlet boundary data are assumed on the whole of ∂Ω and, г, the boundary and the initial data are chosen so that the exact solution is u = cos(2πt) sin(40πx) cos(30πy). The results are plotted on the left of the top row in Figures 3, 4 and 5 for the Galerkin-in-space scheme for errors in u(T), ∇ u(T) and w(T), with the estimated convergence rates tabulated on the right (Notice that a zero value of L2 error for w is reported for the CN scheme when N = 2 — this is no more than a numerical anomaly resulting from exact boundary data and a very coarse mesh.) These are based on the computed errors for N (first column) and for 2N (the results for N = 4 use those for N = 2, the row for which is not shown). The middle row of the figure shows the result of computing with one degree higher polynomials in time than in space, whereas the bottom row shows the result of using one degree higher in space than in time. The CN results are of course unaffected in the middle row. Analogous results for the spectral element scheme in the case where equal degrees are used have also been obtained and are contained in 6.

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