Local refinement techniques for elliptic problems on cell-centered grids. I. Error analysis

Ewing, R. E.; Lazarov, R. D.; Vassilevski, P. S.

doi:10.1090/S0025-5718-1991-1066831-5

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Local refinement techniques for elliptic problems on cell-centered grids. I. Error analysis

Authors: R. E. Ewing, R. D. Lazarov and P. S. Vassilevski
Journal: Math. Comp. 56 (1991), 437-461
MSC: Primary 65N06; Secondary 65N15, 65N50
MathSciNet review: 1066831
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Abstract: A finite difference technique on rectangular cell-centered grids with local refinement is proposed in order to derive discretizations of second-order elliptic equations of divergence type approximating the so-called balance equation. Error estimates in a discrete ${H^1}$ -norm are derived of order ${h^{1/2}}$ for a simple symmetric scheme, and of order ${h^{3/2}}$ for both a nonsymmetric and a more accurate symmetric one, provided that the solution belongs to ${H^{1 + \alpha }}$ for $\alpha > \frac{1}{2}$ and $\alpha > \frac{3}{2}$ , respectively.

[1] Robert A. Adams, Sobolev spaces, Academic Press [A subsidiary of Harcourt Brace Jovanovich, Publishers], New York-London, 1975. Pure and Applied Mathematics, Vol. 65. MR 0450957 (56 #9247)
[2] O. Axelsson, A generalized conjugate gradient, least square method, Numer. Math. 51 (1987), no. 2, 209–227. MR 890033 (88e:65032), http://dx.doi.org/10.1007/BF01396750
[3] O. Axelsson and V. A. Barker, Finite element solution of boundary value problems, Computer Science and Applied Mathematics, Academic Press Inc., Orlando, FL, 1984. Theory and computation. 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), 149-159.
[6] J. H. Bramble and S. R. Hilbert, Bounds for a class of linear functionals with applications to Hermite interpolation, Numer. Math. 16 (1970/1971), 362–369. MR 0290524 (44 #7704)
[7] R. E. Ewing, Efficient adaptive procedures for fluid flow applications, Comput. Methods Appl. Mech. Engrg. 55 (1986), 89-103.
[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., Supra-convergent schemes on irregular grids, Math. Comp. 47 (1986), no. 176, 537–554. MR 856701 (88b:65082), http://dx.doi.org/10.1090/S0025-5718-1986-0856701-5
[10] Thomas A. Manteuffel and Andrew B. White Jr., The numerical solution of second-order boundary value problems on nonuniform meshes, Math. Comp. 47 (1986), no. 176, 511–535, S53–S55. MR 856700 (87m:65116), http://dx.doi.org/10.1090/S0025-5718-1986-0856700-3
[11] S. McCormick, Fast adaptive composite grid (FAC) methods: theory for the variational case, Defect correction methods (Oberwolfach, 1983) Comput. Suppl., vol. 5, Springer, Vienna, 1984, pp. 115–121. MR 782693, http://dx.doi.org/10.1007/978-3-7091-7023-6_7
[12] S. McCormick and J. Thomas, The fast adaptive composite grid (FAC) method for elliptic equations, Math. Comp. 46 (1986), no. 174, 439–456. MR 829618 (87e:65070), http://dx.doi.org/10.1090/S0025-5718-1986-0829618-X
[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 sub-gridding in simulations using a composite grid, Paper SPE 13527, presented at the SPE 1985 Reservoir Simulation Symposium, Dallas, February 1985.
[15] A. A. Samarskiĭ, Homogeneous difference schemes on non-uniform grids for equations of parabolic type, Ž. Vyčisl. Mat. i Mat. Fiz. 3 (1963), 266–298 (Russian). MR 0162366 (28 #5565)
[16] A. A. \cyr{S}amarskiĭ, Vvedenie v teoriyu raznostnykh skhem, Izdat. “Nauka”, Moscow, 1971 (Russian). MR 0347102 (49 #11822)
[17] -, Local one dimensional difference schemes on non-uniform nets, U.S.S.R. Comput. Math. and Math. Phys. 3 (1963), 572-619.
[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 non-uniform nets, U.S.S.R. Comput. Math. and Math. Phys. 2 (1962), 927-953.
[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] Alan Weiser and Mary Fanett Wheeler, On convergence of block-centered finite differences for elliptic problems, SIAM J. Numer. Anal. 25 (1988), no. 2, 351–375. MR 933730 (89m:65094), http://dx.doi.org/10.1137/0725025

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