Axelsson, Owe; Karátson, János Harmonic averages, exact difference schemes and local Green’s functions in variable coefficient PDE problems. (English) Zbl 1273.65107 Cent. Eur. J. Math. 11, No. 8, 1441-1457 (2013). Summary: 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. Cited in 1 Document MSC: 65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs 35L02 First-order hyperbolic equations 35K20 Initial-boundary value problems for second-order parabolic equations 35B25 Singular perturbations in context of PDEs 65M80 Fundamental solutions, Green’s function methods, etc. for initial value and initial-boundary value problems involving PDEs 65M25 Numerical aspects of the method of characteristics for initial value and initial-boundary value problems involving PDEs 65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs 35J25 Boundary value problems for second-order elliptic equations 65M15 Error bounds for initial value and initial-boundary value problems involving PDEs Keywords: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 PDFBibTeX XMLCite \textit{O. Axelsson} and \textit{J. Karátson}, Cent. Eur. J. Math. 11, No. 8, 1441--1457 (2013; Zbl 1273.65107) Full Text: DOI Link References: [1] Axelsson O., Stability and error estimates of Galerkin finite element approximations for convection-diffusion equations, IMA J. Numer. Anal., 1981, 1(3), 329-345 http://dx.doi.org/10.1093/imanum/1.3.329; · Zbl 0508.76069 [2] Axelsson O., Finite difference methods, In: Encyclopedia of Computational Mechanics, 1, John Wiley & Sons, Chichester, 2004; [3] Axelsson O., Glushkov E., Glushkova N., The local Green’s function method in singularly perturbed convection-diffusion problems, Math. Comp., 2009, 78(265), 153-170 http://dx.doi.org/10.1090/S0025-5718-08-02161-3; · Zbl 1198.65058 [4] Axelsson O., Gololobov S.V., A combined method of local Green’s functions and central difference method for singularly perturbed convection-diffusion problems, J. Comput. Appl. Math., 2003, 161(2), 245-257 http://dx.doi.org/10.1016/j.cam.2003.08.005; · Zbl 1037.65104 [5] Axelsson O., Karátson J., Mesh independent superlinear PCG rates via compact-equivalent operators, SIAM J. Numer. Anal., 2007, 45(4), 1495-1516 http://dx.doi.org/10.1137/06066391X; · Zbl 1151.65081 [6] Babuška I., Caloz G., Osborn E., Special finite element methods for a class of second order elliptic problems with rough coefficients, SIAM J. Numer. Anal., 1994, 31(4), 945-981 http://dx.doi.org/10.1137/0731051; · Zbl 0807.65114 [7] Efendiev Y., Hou T., Strinopoulos T., Multiscale simulations of porous media flows in flow-based coordinate system, Comput. Geosci., 2008, 12(3), 257-272 http://dx.doi.org/10.1007/s10596-007-9073-7; · Zbl 1155.76050 [8] Hemker P.W., A Numerical Study of Stiff Two-Point Boundary Problems, Math. Centre Tracts, 80, Mathematisch Centrum, Amsterdam, 1977; · Zbl 0426.65043 [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; · Zbl 0484.65065 [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; · Zbl 0377.65045 [11] Matus P., Irkhin V., Lapinska-Chrzczonowicz M., Exact difference schemes for time-dependent problems, Comput. Methods Appl. Math., 2005, 5(4), 422-448; · Zbl 1082.65076 [12] Samarskii A.A., The Theory of Difference Schemes, Monogr. Textbooks Pure Appl. Math., 240, Marcel Dekker, New York, 2001; · Zbl 0971.65076 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.