{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,22]],"date-time":"2026-04-22T14:15:38Z","timestamp":1776867338142,"version":"3.51.2"},"reference-count":28,"publisher":"Springer Science and Business Media LLC","issue":"3","license":[{"start":{"date-parts":[[1990,9,1]],"date-time":"1990-09-01T00:00:00Z","timestamp":652147200000},"content-version":"tdm","delay-in-days":0,"URL":"http:\/\/www.springer.com\/tdm"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Computing"],"published-print":{"date-parts":[[1990,9]]},"DOI":"10.1007\/bf02250633","type":"journal-article","created":{"date-parts":[[2005,11,22]],"date-time":"2005-11-22T14:29:06Z","timestamp":1132669746000},"page":"193-215","source":"Crossref","is-referenced-by-count":22,"title":["Finite difference schemes on grids with local refinement in time and space for parabolic problems I. Derivation, stability, and error analysis","Differenz-Schemata f\u00fcr parabolische Probleme auf Gittern mit Lokaler Verdichtung in Zeit und Raum"],"prefix":"10.1007","volume":"45","author":[{"given":"R. E.","family":"Ewing","sequence":"first","affiliation":[]},{"given":"R. D.","family":"Lazarov","sequence":"additional","affiliation":[]},{"given":"P. S.","family":"Vassilevski","sequence":"additional","affiliation":[]}],"member":"297","reference":[{"key":"BF02250633_CR1","volume-title":"Sobolev spaces","author":"R. A. Adams","year":"1975","unstructured":"Adams, R. A.: Sobolev spaces. New York: Academic Press 1975."},{"key":"BF02250633_CR2","volume-title":"Petroleum Reservior Simulation","author":"K. Aziz","year":"1979","unstructured":"Aziz, K., Settari, A.: Petroleum Reservior Simulation, London: Applied Science Publishers 1979."},{"key":"BF02250633_CR3","doi-asserted-by":"crossref","first-page":"1597","DOI":"10.1002\/nme.1620121010","volume":"12","author":"I. Babuska","year":"1978","unstructured":"Babuska, I., Rheinboldt, W. C.: A posteriori error estimates for the finite element method. Int. J. Numer. Meth.,12 (1978), 1597\u20131615.","journal-title":"Int. J. Numer. Meth."},{"issue":"4","key":"BF02250633_CR4","doi-asserted-by":"crossref","first-page":"736","DOI":"10.1137\/0715049","volume":"15","author":"I. Babuska","year":"1978","unstructured":"Babuska, I., Rheiboldt, W. C.: Error estimates for adaptive finite element computations. SIAM J. Numer. Anal.,15 (4) (1978), 736\u2013754.","journal-title":"SIAM J. Numer. Anal."},{"key":"BF02250633_CR5","doi-asserted-by":"crossref","first-page":"149","DOI":"10.1016\/0045-7825(88)90122-3","volume":"67","author":"J. H. Bramble","year":"1988","unstructured":"Bramble, J. H., Ewing, R. E., Pasciak, J. E., Schatz, A. H.: A preconditioning technique for the efficient solution of problems with local grid refinement. Comp. Meth. Appl. Eng.,67 (1988), 149\u2013159.","journal-title":"Comp. Meth. Appl. Eng."},{"key":"BF02250633_CR6","doi-asserted-by":"crossref","first-page":"29","DOI":"10.1016\/0045-7825(83)90051-8","volume":"41","author":"A. R. Diaz","year":"1983","unstructured":"Diaz, A. R., Kikuchi, N., Taylor, J. I.: A method of grid optimization for finite element methods. Comp. Meth. Appl. Mech. Eng.,41 (1983), 29\u201345.","journal-title":"Comp. Meth. Appl. Mech. Eng."},{"key":"BF02250633_CR7","doi-asserted-by":"crossref","first-page":"89","DOI":"10.1016\/0045-7825(86)90087-3","volume":"55","author":"R. E. Ewing","year":"1986","unstructured":"Ewing, R. E.: Efficient adaptive procedures for fluid flow applications. Comp. Meth. Appl. Mech. Eng.,55 (1986), 89\u2013103.","journal-title":"Comp. Meth. Appl. Mech. Eng."},{"key":"BF02250633_CR8","unstructured":"Ewing, R. E., Lazarov, R. D., Vassilevski, P. S.: Local refinement techniques for elliptic problems on cell-centered grids. Preprint #1988-16, Institute for Scientific Computation, University of Wyoming, 1988."},{"key":"BF02250633_CR9","doi-asserted-by":"crossref","first-page":"537","DOI":"10.1090\/S0025-5718-1986-0856701-5","volume":"47","author":"H. O. Kreiss","year":"1986","unstructured":"Kreiss, H. O., Manteuffel, T. A., Swartz, B., Wendroff, B., White, A. B. Jr.: Superconvergence schemes on irregular grid. Math. Comput.,47 (1986), 537\u2013554.","journal-title":"Math. Comput."},{"key":"BF02250633_CR10","doi-asserted-by":"crossref","first-page":"511","DOI":"10.1090\/S0025-5718-1986-0856700-3","volume":"47","author":"T. A. Manteuffel","year":"1986","unstructured":"Manteuffel, T. A., White, A. B. Jr.: The numerical solution of second order boundary value problems on nonuniform meshes. Math. Comput.,47 (1986), 511\u2013535.","journal-title":"Math. Comput."},{"key":"BF02250633_CR11","first-page":"115","volume":"Suppl., 5","author":"S. McCormick","year":"1984","unstructured":"McCormick, S.: Fast adaptive composite grid (FAC) methods: theory for the variational case. Comput. Suppl.,5 (1984), 115\u2013121.","journal-title":"Comput."},{"key":"BF02250633_CR12","doi-asserted-by":"crossref","first-page":"439","DOI":"10.1090\/S0025-5718-1986-0829618-X","volume":"46","author":"S. McCormick","year":"1986","unstructured":"McCormick, S., Thomas, J.: The fast adaptive composite grid (FAC) method for elliptic equations. Math. Comput.,46 (1986), 439\u2013456.","journal-title":"Math. Comput."},{"key":"BF02250633_CR13","first-page":"249","volume-title":"Accuracy Estimates and Adaptive Refinement in Finite Element Computations","author":"J. T. Oden","year":"1986","unstructured":"Oden, J. T., Demkowicz, L., Strouboulis, T., Devloo, P.: Adaptive methods for problems in solid and fluid mechanics. In Accuracy Estimates and Adaptive Refinement in Finite Element Computations, I. Babuska et al., eds., Wiley, New York, 1986, 249\u2013280."},{"key":"BF02250633_CR14","unstructured":"Pedrosa, O. A. Jr.: Use of Hybrid Grid in Reservoir Simulation, Ph.D. Thesis, Stanford University, 1984."},{"key":"BF02250633_CR15","doi-asserted-by":"crossref","unstructured":"Quandalle, P., Besset, P.: 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.","DOI":"10.2118\/13527-MS"},{"key":"BF02250633_CR16","doi-asserted-by":"crossref","first-page":"351","DOI":"10.1016\/0041-5553(63)90025-9","volume":"3","author":"A. A. Samarskii","year":"1963","unstructured":"Samarskii, A. A.: Homogeneous difference schemes on non-uniform nets for equations of parabolic type. USSR Comput. Math. and Math. Phys.,3 (1963), 351\u2013393.","journal-title":"USSR Comput. Math. and Math. Phys."},{"key":"BF02250633_CR17","first-page":"468","volume-title":"Introduction to Theory of Difference Schemes","author":"A. A. Samarskii","year":"1971","unstructured":"Samarskii, A. A.: Introduction to Theory of Difference Schemes, p. 468, Moscow: Nauka 1971 (Russian)."},{"key":"BF02250633_CR18","doi-asserted-by":"crossref","first-page":"572","DOI":"10.1016\/0041-5553(63)90290-8","volume":"3","author":"A. A. Samarskii","year":"1963","unstructured":"Samarskii, A. A.: Local one dimensional difference schemes on non-uniform nets. USSR Comput. Math. and Math. Phys.,3 (1963), 572\u2013619.","journal-title":"USSR Comput. Math. and Math. Phys."},{"key":"BF02250633_CR19","first-page":"296","volume-title":"Difference Schemes for Differential Equations having Generalized Solutions","author":"A. A. Samarskii","year":"1987","unstructured":"Samarskii, A. A., Lazarov, R. D., Makarov, V. L.: Difference Schemes for Differential Equations having Generalized Solutions, p. 296, Moscow: Vysshaya Shkola Publishers, 1987 (Russian)."},{"key":"BF02250633_CR20","doi-asserted-by":"crossref","first-page":"927","DOI":"10.1016\/0041-5553(63)90505-6","volume":"2","author":"A. N. Tikhonov","year":"1962","unstructured":"Tikhonov, A. N., Samarskii, A. A.: Homogeneous difference schemes on nonuniform nets. USSR Comput. Math. Phys.,2 (1962), 927\u2013953.","journal-title":"USSR Comput. Math. Phys."},{"key":"BF02250633_CR21","doi-asserted-by":"crossref","first-page":"351","DOI":"10.1137\/0725025","volume":"25","author":"A. Weiser","year":"1988","unstructured":"Weiser, A., Wheeler, M. F.: On convergence of block-centered finite-differences for elliptic problems. SIAM J. Numer. Anal.,25 (1988), 351\u2013375.","journal-title":"SIAM J. Numer. Anal."},{"key":"BF02250633_CR22","volume-title":"Adaptive finite element methods for linear parabolic problems. II. A priori error estimates ofL \u221e (L 2)","author":"K. Eriksson","year":"1988","unstructured":"Eriksson, K.: Adaptive finite element methods for linear parabolic problems. II. A priori error estimates ofL \u221e (L 2). Technical Report, Department of Mathematics, Chalmers University of Technology, Goteborg, 1988."},{"key":"BF02250633_CR23","volume-title":"Adaptive finite element methods for linear parabolic problems. III. A priori error estimates inL \u221e (L \u221e )","author":"K. Eriksson","year":"1988","unstructured":"Eriksson, K.: Adaptive finite element methods for linear parabolic problems. III. A priori error estimates inL \u221e (L \u221e ). Technical Report, Department of Mathematics, Chalmers University of Technology, Goteborg, 1988."},{"key":"BF02250633_CR24","unstructured":"Eriksson, K., Johnson, C.: Adaptive finite element methods for parabolic problems IV. Time steps variable in space (to appear)."},{"key":"BF02250633_CR25","unstructured":"Herox, M. A., Thomas, J. W.: TDFAC: A composite grid method for time dependent problems. Proceedings Fourth Copper Mountain Conference on Multigrid Methods, 1989."},{"key":"BF02250633_CR26","unstructured":"Ewing, R. E., Lazarov, R. D., Pasciak, J. E., Vassilevski, P. S.: Finite element methods for parabolic problems with time steps variable in space. Preprint #1989-05, Institute for Scientific Computation, University of Wyoming, 1989."},{"key":"BF02250633_CR27","unstructured":"Matus, P. P.: On a class of difference schemes for transient boundary value problems. Inst. Math., Minsk Acad. Sci. BSSR, Preprint23, 1989."},{"key":"BF02250633_CR28","unstructured":"Ewing, R. E., Lazarov, R. D., Vassilevski, P. S.: Finite difference schemes on grids with local refinement in time and space for parabolic problems. II: Optimal order two-grid iterative methods. Proceedings, Sixth GAMM Seminar on Parallel Methods for PDEs, Kiel, January 22\u201324, 1990, to appear in Notes in Numerical Fluid Mechanics, Vieweg-Braunsweig, 1990."}],"container-title":["Computing"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/BF02250633.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/article\/10.1007\/BF02250633\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/BF02250633","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,5,15]],"date-time":"2019-05-15T17:48:33Z","timestamp":1557942513000},"score":1,"resource":{"primary":{"URL":"http:\/\/link.springer.com\/10.1007\/BF02250633"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1990,9]]},"references-count":28,"journal-issue":{"issue":"3","published-print":{"date-parts":[[1990,9]]}},"alternative-id":["BF02250633"],"URL":"https:\/\/doi.org\/10.1007\/bf02250633","relation":{},"ISSN":["0010-485X","1436-5057"],"issn-type":[{"value":"0010-485X","type":"print"},{"value":"1436-5057","type":"electronic"}],"subject":[],"published":{"date-parts":[[1990,9]]}}}