On a Galerkin Discretization of 4th order in Space and Time Applied to the Heat Equation
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@Article{IJNAMB-4-353,
author = {},
title = {On a Galerkin Discretization of 4th order in Space and Time Applied to the Heat Equation},
journal = {International Journal of Numerical Analysis Modeling Series B},
year = {2013},
volume = {4},
number = {4},
pages = {353--371},
abstract = {We present a new time discretization scheme based on the continuous Galerkin Petrov method of polynomial order 3 (cGP(3)-method) which is combined with a reduced numerical time
integration (3-point Gauß-Lobatto formula). The solution of the new approach can be computed
from the solution of the lower order cGP(2)-method, which requires to solve a coupled 2 × 2 block
system on each time interval, followed by a simple post-processing step, such that we get the higher
accuracy of 4th order in time in the standard L²-norm with nearly the cost of the cGP(2)-method.
Moreover, the difference of both solutions can be used as an indicator for the approximation error
in time. For the approximation in space we use the nonparametric \tilde{Q}_3-element which belongs
to a family of recently derived higher order nonconforming finite element spaces and leads to an
approximation error in space of order 4, too, in the L²-norm. The expected optimal accuracy of
the full discretization error in the L²-norm of 4th order in space and time is confirmed by several
numerical tests. We discuss implementation aspects of the time discretization as well as efficient
multigrid methods for solving the resulting block systems which lead to convergence rates being
almost independent of the mesh size and the time step. In our numerical experiments we compare
different higher order spatial and temporal discretization approaches with respect to accuracy and
computational cost.},
issn = {},
doi = {https://doi.org/},
url = {http://global-sci.org/intro/article_detail/ijnamb/262.html}
}
TY - JOUR
T1 - On a Galerkin Discretization of 4th order in Space and Time Applied to the Heat Equation
JO - International Journal of Numerical Analysis Modeling Series B
VL - 4
SP - 353
EP - 371
PY - 2013
DA - 2013/04
SN - 4
DO - http://doi.org/
UR - https://global-sci.org/intro/article_detail/ijnamb/262.html
KW - continuous Galerkin-Petrov method
KW - nonconforming FEM
KW - heat equation
KW - multigrid method
AB - We present a new time discretization scheme based on the continuous Galerkin Petrov method of polynomial order 3 (cGP(3)-method) which is combined with a reduced numerical time
integration (3-point Gauß-Lobatto formula). The solution of the new approach can be computed
from the solution of the lower order cGP(2)-method, which requires to solve a coupled 2 × 2 block
system on each time interval, followed by a simple post-processing step, such that we get the higher
accuracy of 4th order in time in the standard L²-norm with nearly the cost of the cGP(2)-method.
Moreover, the difference of both solutions can be used as an indicator for the approximation error
in time. For the approximation in space we use the nonparametric \tilde{Q}_3-element which belongs
to a family of recently derived higher order nonconforming finite element spaces and leads to an
approximation error in space of order 4, too, in the L²-norm. The expected optimal accuracy of
the full discretization error in the L²-norm of 4th order in space and time is confirmed by several
numerical tests. We discuss implementation aspects of the time discretization as well as efficient
multigrid methods for solving the resulting block systems which lead to convergence rates being
almost independent of the mesh size and the time step. In our numerical experiments we compare
different higher order spatial and temporal discretization approaches with respect to accuracy and
computational cost.
SHAFQAT HUSSAIN, FRIEDHELM SCHIEWECK, STEFAN TUREK, AND PETER ZAJAC. (1970). On a Galerkin Discretization of 4th order in Space and Time Applied to the Heat Equation.
International Journal of Numerical Analysis Modeling Series B. 4 (4).
353-371.
doi:
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