Optimal Order Convergence Implies Numerical Smoothness.
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@Article{IJNAMB-5-357,
author = {SO–HSIANG CHOU},
title = {Optimal Order Convergence Implies Numerical Smoothness.},
journal = {International Journal of Numerical Analysis Modeling Series B},
year = {2014},
volume = {5},
number = {4},
pages = {357--373},
abstract = {It is natural to expect the following loosely stated approximation principle to hold: a
numerical approximation solution should be in some sense as smooth as its target exact solution
in order to have optimal convergence. For piecewise polynomials, that means we have to at least
maintain numerical smoothness in the interiors as well as across the interfaces of cells or elements.
In this paper we give clear definitions of numerical smoothness that address the across-interface
smoothness in terms of scaled jumps in derivatives [9] and the interior numerical smoothness in
terms of differences in derivative values. Furthermore, we prove rigorously that the principle can
be simply stated as numerical smoothness is necessary for optimal order convergence. It is valid
on quasi-uniform meshes by triangles and quadrilaterals in two dimensions and by tetrahedrons
and hexahedrons in three dimensions. With this validation we can justify, among other things,
incorporation of this principle in creating adaptive numerical approximation for the solution of
PDEs or ODEs, especially in designing proper smoothness indicators or detecting potential nonconvergence
and instability.},
issn = {},
doi = {https://doi.org/},
url = {http://global-sci.org/intro/article_detail/ijnamb/240.html}
}
TY - JOUR
T1 - Optimal Order Convergence Implies Numerical Smoothness.
AU - SO–HSIANG CHOU
JO - International Journal of Numerical Analysis Modeling Series B
VL - 4
SP - 357
EP - 373
PY - 2014
DA - 2014/05
SN - 5
DO - http://doi.org/
UR - https://global-sci.org/intro/article_detail/ijnamb/240.html
KW - Adaptive algorithm
KW - discontinuous Galerkin
KW - numerical smoothness
KW - optimal order convergence
AB - It is natural to expect the following loosely stated approximation principle to hold: a
numerical approximation solution should be in some sense as smooth as its target exact solution
in order to have optimal convergence. For piecewise polynomials, that means we have to at least
maintain numerical smoothness in the interiors as well as across the interfaces of cells or elements.
In this paper we give clear definitions of numerical smoothness that address the across-interface
smoothness in terms of scaled jumps in derivatives [9] and the interior numerical smoothness in
terms of differences in derivative values. Furthermore, we prove rigorously that the principle can
be simply stated as numerical smoothness is necessary for optimal order convergence. It is valid
on quasi-uniform meshes by triangles and quadrilaterals in two dimensions and by tetrahedrons
and hexahedrons in three dimensions. With this validation we can justify, among other things,
incorporation of this principle in creating adaptive numerical approximation for the solution of
PDEs or ODEs, especially in designing proper smoothness indicators or detecting potential nonconvergence
and instability.
SO–HSIANG CHOU. (2014). Optimal Order Convergence Implies Numerical Smoothness..
International Journal of Numerical Analysis Modeling Series B. 5 (4).
357-373.
doi:
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