Local refinement techniques for elliptic problems on cellcentered grids. I. Error analysis
Authors:
R. E. Ewing, R. D. Lazarov and P. S. Vassilevski
Journal:
Math. Comp. 56 (1991), 437461
MSC:
Primary 65N06; Secondary 65N15, 65N50
MathSciNet review:
1066831
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Abstract: A finite difference technique on rectangular cellcentered grids with local refinement is proposed in order to derive discretizations of secondorder elliptic equations of divergence type approximating the socalled balance equation. Error estimates in a discrete norm are derived of order for a simple symmetric scheme, and of order for both a nonsymmetric and a more accurate symmetric one, provided that the solution belongs to for and , respectively.
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 R. A. Adams, Sobolev spaces, Academic Press, New York, 1975. MR 0450957 (56:9247)
 [2]
 O. Axelsson, A generalized conjugate gradient, least square method, Numer. Math. 51 (1987), 209227. MR 890033 (88e:65032)
 [3]
 O. Axelsson and V. A. Baker, Finite element solutions of boundary value problems, theory and computation, Academic Press, Orlando, 1984. MR 758437 (85m:65116)
 [4]
 K. Aziz and A. Settari, Petroleum reservoir simulation, Applied Science Publishers, London, 1979.
 [5]
 J. H. Bramble, R. E. Ewing, J. E. Pasciak, and A. H. Schatz, A preconditioning technique for the efficient solution of problems with local grid refinement, Comput. Methods Appl. Mech. Engrg. 67 (1988), 149159.
 [6]
 J. H. Bramble and S. R. Hilbert, Bounds for a class of linear functionals with application to Hermite interpolation, Numer. Math. 16 (1971), 362369. MR 0290524 (44:7704)
 [7]
 R. E. Ewing, Efficient adaptive procedures for fluid flow applications, Comput. Methods Appl. Mech. Engrg. 55 (1986), 89103.
 [8]
 R. E. Ewing and R. D. Lazarov, Adaptive local grid refinement, Paper SPE 17806, presented at the SPE Rocky Mountain Regional Meeting, Casper, May 1988.
 [9]
 H. O. Kreiss, T. A. Manteuffel, B. Swartz, B. Wendroff, and A. B. White, Jr., Superconvergent schemes on irregular grids, Math. Comp. 47 (1986), 537554. MR 856701 (88b:65082)
 [10]
 T. A. Manteuffel and A. B. White, Jr., The numerical solution of secondorder boundary value problems on nonuniform meshes, Math. Comp. 47 (1986), 511535. MR 856700 (87m:65116)
 [11]
 S. McCormick, Fast adaptive composite grid (FAC) methods: Theory for the variational case, Comput. Suppl. 5 (1984), 115121. MR 782693
 [12]
 S. McCormick and J. Thomas, The fast adaptive composite grid (FAC) method for elliptic equations, Math. Comp. 46 (1986), 439456. MR 829618 (87e:65070)
 [13]
 O. A. Pedrosa, Jr., Use of hybrid grid in reservoir simulation, Ph. D. Thesis, Stanford University, 1984.
 [14]
 P. Quandalle and P. Besset, Reduction of grid effects due to local subgridding in simulations using a composite grid, Paper SPE 13527, presented at the SPE 1985 Reservoir Simulation Symposium, Dallas, February 1985.
 [15]
 A. A. Samarskii, Homogeneous difference schemes on nonuniform nets for equations of parabolic type, U.S.S.R. Comput. Math. and Math. Phys. 3 (1963), 351393. MR 0162366 (28:5565)
 [16]
 , Introduction to the theory of difference schemes, Nauka, Moskow, 1971. (Russian) MR 0347102 (49:11822)
 [17]
 , Local one dimensional difference schemes on nonuniform nets, U.S.S.R. Comput. Math. and Math. Phys. 3 (1963), 572619.
 [18]
 A. A. Samarskii, R. D. Lazarov, and V. L. Makarov, Difference schemes for differential equations having generalized solutions, Vysshaya Shkola, Moskow, USSR, 1987. (Russian)
 [19]
 A. N. Tikhonov and A. A. Samarskii, Homogeneous difference schemes on nonuniform nets, U.S.S.R. Comput. Math. and Math. Phys. 2 (1962), 927953.
 [20]
 D. U. von Rosenberg, Local grid refinement for finite difference methods, Paper SPE 10974, presented at the 57th Annual Fall Technical Conference, New Orleans, September 1982.
 [21]
 A. Weiser and M. F. Wheeler, On convergence of blockcentered finite differences for elliptic problems, SIAM J. Numer. Anal. 25 (1988), 351375. MR 933730 (89m:65094)
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00255718199110668315
PII:
S 00255718(1991)10668315
Keywords:
Cellcentered grid,
local refinement,
error estimates,
elliptic problems of divergence type
Article copyright:
© Copyright 1991
American Mathematical Society
