Harmonic averages, exact difference schemes and local Green’s functions in variable coefficient PDE problems

Owe Axelsson; János Karátson

Open Mathematics (2013)

  • Volume: 11, Issue: 8, page 1441-1457
  • ISSN: 2391-5455

Abstract

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A brief survey is given to show that harmonic averages enter in a natural way in the numerical solution of various variable coefficient problems, such as in elliptic and transport equations, also of singular perturbation types. Local Green’s functions used as test functions in the Petrov-Galerkin finite element method combined with harmonic averages can be very efficient and are related to exact difference schemes.

How to cite

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Owe Axelsson, and János Karátson. "Harmonic averages, exact difference schemes and local Green’s functions in variable coefficient PDE problems." Open Mathematics 11.8 (2013): 1441-1457. <http://eudml.org/doc/269454>.

@article{OweAxelsson2013,
abstract = {A brief survey is given to show that harmonic averages enter in a natural way in the numerical solution of various variable coefficient problems, such as in elliptic and transport equations, also of singular perturbation types. Local Green’s functions used as test functions in the Petrov-Galerkin finite element method combined with harmonic averages can be very efficient and are related to exact difference schemes.},
author = {Owe Axelsson, János Karátson},
journal = {Open Mathematics},
keywords = {Variable coefficients; Harmonic averages; Singular perturbation; Local Green’s functions; Exact difference schemes; variable coefficients; harmonic averages; singular perturbation; local Green's functions; exact difference schemes; convection-reaction equation; error estimates; Darcy flow; method of characteristics; transport equations; Petrov-Galerkin finite element method},
language = {eng},
number = {8},
pages = {1441-1457},
title = {Harmonic averages, exact difference schemes and local Green’s functions in variable coefficient PDE problems},
url = {http://eudml.org/doc/269454},
volume = {11},
year = {2013},
}

TY - JOUR
AU - Owe Axelsson
AU - János Karátson
TI - Harmonic averages, exact difference schemes and local Green’s functions in variable coefficient PDE problems
JO - Open Mathematics
PY - 2013
VL - 11
IS - 8
SP - 1441
EP - 1457
AB - A brief survey is given to show that harmonic averages enter in a natural way in the numerical solution of various variable coefficient problems, such as in elliptic and transport equations, also of singular perturbation types. Local Green’s functions used as test functions in the Petrov-Galerkin finite element method combined with harmonic averages can be very efficient and are related to exact difference schemes.
LA - eng
KW - Variable coefficients; Harmonic averages; Singular perturbation; Local Green’s functions; Exact difference schemes; variable coefficients; harmonic averages; singular perturbation; local Green's functions; exact difference schemes; convection-reaction equation; error estimates; Darcy flow; method of characteristics; transport equations; Petrov-Galerkin finite element method
UR - http://eudml.org/doc/269454
ER -

References

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  8. [8] Hemker P.W., A Numerical Study of Stiff Two-Point Boundary Problems, Math. Centre Tracts, 80, Mathematisch Centrum, Amsterdam, 1977 Zbl0426.65043
  9. [9] Houstis E.N., Rice J.R., High order methods for elliptic partial differential equations with singularities, Internat. J. Numer. Methods Engrg., 1982, 18(5), 737–754 http://dx.doi.org/10.1002/nme.1620180509 Zbl0484.65065
  10. [10] Lynch R.E., Rice J.R., High accuracy finite difference approximation to solutions of elliptic partial differential equations, Proc. Nat. Acad. Sci. U.S.A., 1978, 75(6), 2541–2544 http://dx.doi.org/10.1073/pnas.75.6.2541 Zbl0377.65045
  11. [11] Matus P., Irkhin V., Lapinska-Chrzczonowicz M., Exact difference schemes for time-dependent problems, Comput. Methods Appl. Math., 2005, 5(4), 422–448 Zbl1082.65076
  12. [12] Samarskii A.A., The Theory of Difference Schemes, Monogr. Textbooks Pure Appl. Math., 240, Marcel Dekker, New York, 2001 Zbl0971.65076

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