{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,9,29]],"date-time":"2026-09-29T06:21:58Z","timestamp":1790662918228,"version":"4.1.0"},"reference-count":34,"publisher":"Society for Industrial & Applied Mathematics (SIAM)","issue":"3","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["SIAM J. Numer. Anal."],"published-print":{"date-parts":[[2008,1]]},"abstract":"<jats:p>In this paper, we analyze a multiscale operator splitting method for solving systems of ordinary differential equations such as those that result upon space discretization of a reaction-diffusion equation. Our goal is to analyze and accurately estimate the error of the numerical solution, including the effects of any instabilities that can result from multiscale operator splitting. We present both an a priori error analysis and a new type of hybrid a priori\u2013a posteriori error analysis for an operator splitting discontinuous Galerkin finite element method. Both analyses clearly distinguish between the effects of the operator splitting and the discretization of each component of the decomposed problem. The hybrid analysis has the form of a computable a posteriori leading order expression and a provably higher order a priori expression. The hybrid analysis takes into account the fact that the adjoint problems for the original problem and a multiscale operator splitting discretization differ in significant ways. In particular, this provides the means to monitor global instabilities that can arise from operator splitting.<\/jats:p>","DOI":"10.1137\/07068237x","type":"journal-article","created":{"date-parts":[[2008,3,7]],"date-time":"2008-03-07T18:06:03Z","timestamp":1204913163000},"page":"1116-1146","source":"Crossref","is-referenced-by-count":43,"title":["An A Posteriori\u2013A Priori Analysis of Multiscale Operator Splitting"],"prefix":"10.1137","volume":"46","author":[{"given":"D.","family":"Estep","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"V.","family":"Ginting","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"D.","family":"Ropp","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"J. N.","family":"Shadid","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"S.","family":"Tavener","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2008,3,7]]},"reference":[{"key":"R1","doi-asserted-by":"publisher","DOI":"10.1002\/cpa.3160430804"},{"key":"R2","unstructured":"V. Carey, D. Estep, and S. Tavener,\n                      A posteriori analysis and adaptive error control for multiscale operator decomposition solution of elliptic systems\n                      I:\n                      Triangular systems\n                      , SIAM J. Numer. Anal., submitted (2006)."},{"key":"R3","doi-asserted-by":"publisher","DOI":"10.1002\/cpa.3160420502"},{"key":"R4","doi-asserted-by":"crossref","unstructured":"P. G. Ciarlet,\n                      The Finite Element Method for Elliptic Problems\n                      , Stud. Math. Appl. 4, North\u2013Holland, Amsterdam, 1978.","DOI":"10.1115\/1.3424474"},{"key":"R5","unstructured":"C. N. Dawson and M. F. Wheeler,\n                      Time-splitting methods for advection-diffusion-reaction equations arising in contaminant transport\n                      , in ICIAM 91 (Washington, DC, 1991), SIAM, Philadelphia, 1992, pp. 71\u201382."},{"key":"R6","doi-asserted-by":"publisher","DOI":"10.1090\/S0025-5718-1986-0842129-0"},{"key":"R7","doi-asserted-by":"publisher","DOI":"10.1090\/S0025-5718-1981-0606506-0"},{"key":"R8","doi-asserted-by":"publisher","DOI":"10.1090\/S0025-5718-00-01277-1"},{"key":"R9","unstructured":"S. Eastman,\n                      Analysis and Application of the Nonlinear Power Method\n                      , Ph.D. thesis, Colorado State University, Fort Collins, CO, 2005."},{"key":"R10","doi-asserted-by":"crossref","unstructured":"K. Eriksson, D. Estep, P. Hansbo, and C. Johnson,\n                      Introduction to adaptive methods for differential equations\n                      , in Acta Numerica, 1995, Cambridge University Press, Cambridge, UK, 1995, pp. 105\u2013158.","DOI":"10.1017\/S0962492900002531"},{"key":"R11","unstructured":"K. Eriksson, D. Estep, P. Hansbo, and C. Johnson,\n                      Computational Differential Equations\n                      , Cambridge University Press, Cambridge, UK, 1996."},{"key":"R12","doi-asserted-by":"publisher","DOI":"10.1088\/0951-7715\/7\/5\/011"},{"key":"R13","doi-asserted-by":"publisher","DOI":"10.1137\/0732001"},{"key":"R14","unstructured":"D. Estep,\n                      A Short Course on Duality, Adjoint Operators, Green's Functions, and A Posteriori Error Analysis\n                      , Sandia National Laboratories, Albuquerque, NM, 2004. Notes can be downloaded from http:\/\/math.colostate.edu\/~estep."},{"key":"R15","doi-asserted-by":"publisher","DOI":"10.1051\/m2an\/1994280708151"},{"key":"R16","doi-asserted-by":"publisher","DOI":"10.1090\/S0025-5718-01-01364-3"},{"key":"R17","doi-asserted-by":"crossref","unstructured":"D. Estep, M. G. Larson, and R. D. Williams,\n                      Estimating the error of numerical solutions of systems of reaction-diffusion equations\n                      , Mem. Amer. Math. Soc., 146 (2000).","DOI":"10.1090\/memo\/0696"},{"key":"R18","unstructured":"D. Estep, S. Tavener, and T. Wildey,\n                      A posteriori analysis and improved accuracy for an operator decomposition solution of a conjugate heat transfer problem\n                      , SIAM J. Numer. Anal., to appear."},{"key":"R19","doi-asserted-by":"publisher","DOI":"10.1007\/BF01048791"},{"key":"R20","doi-asserted-by":"crossref","unstructured":"W. Hundsdorfer and J. Verwer,\n                      Numerical Solution of Time-Dependent Advection-Diffusion-Reaction Equations\n                      , Springer Ser. Comput. Math. 33, Springer-Verlag, Berlin, 2003.","DOI":"10.1007\/978-3-662-09017-6"},{"key":"R21","doi-asserted-by":"publisher","DOI":"10.1090\/S0025-5718-1989-0969489-7"},{"key":"R22","doi-asserted-by":"publisher","DOI":"10.1137\/0725051"},{"key":"R23","doi-asserted-by":"crossref","unstructured":"G. I. Marchuk,\n                      On the theory of the splitting-up method\n                      , in Proceedings of the Second Symposium on Numerical Solution of Partial Differential Equations, SVNSPADE, Academic Press, New York, 1970, pp. 469\u2013500.","DOI":"10.1016\/B978-0-12-358502-8.50019-0"},{"key":"R24","unstructured":"G. I. Marchuk, V. I. Agoshkov, and V. P. Shutyaev,\n                      Adjoint Equations and Perturbation Algorithms in Nonlinear Problems\n                      , CRC Press, Boca Raton, FL, 1996."},{"key":"R25","doi-asserted-by":"publisher","DOI":"10.1016\/j.jcp.2003.10.036"},{"key":"R26","doi-asserted-by":"publisher","DOI":"10.1063\/1.1668896"},{"key":"R27","doi-asserted-by":"publisher","DOI":"10.1016\/j.jcp.2004.09.004"},{"key":"R28","doi-asserted-by":"publisher","DOI":"10.1016\/j.jcp.2003.08.033"},{"key":"R29","unstructured":"J. Sandelin,\n                      Global Estimate and Control of Model, Numerical, and Parameter Error\n                      , Ph.D. thesis, Department of Mathematics, Colorado State University, Fort Collins, CO, 2006."},{"key":"R30","doi-asserted-by":"crossref","unstructured":"J. A. Smoller,\n                      Shock Waves and Reaction-Diffusion Equations\n                      , 2nd ed., Springer-Verlag, New York, 1983.","DOI":"10.1007\/978-1-4684-0152-3"},{"key":"R31","doi-asserted-by":"publisher","DOI":"10.1006\/jcph.2000.6495"},{"key":"R32","doi-asserted-by":"publisher","DOI":"10.1137\/0705041"},{"key":"R33","unstructured":"G. Strang and G. J. Fix,\n                      An Analysis of the Finite Element Method\n                      , Prentice\u2013Hall Ser. Automat. Comput., Prentice\u2013Hall, Englewood Cliffs, NJ, 1973."},{"key":"R34","unstructured":"J. G. Verwer and B. Sportisse,\n                      A Note on Operator Splitting Analysis in the Stiff Linear Case\n                      , CWI report MAS-R9801, Amsterdam, The Netherlands, 1998."}],"container-title":["SIAM Journal on Numerical Analysis"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/epubs.siam.org\/doi\/pdf\/10.1137\/07068237X","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T15:13:09Z","timestamp":1787325189000},"score":1,"resource":{"primary":{"URL":"https:\/\/epubs.siam.org\/doi\/10.1137\/07068237X"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2008,1]]},"references-count":34,"journal-issue":{"issue":"3","published-print":{"date-parts":[[2008,1]]}},"alternative-id":["10.1137\/07068237X"],"URL":"https:\/\/doi.org\/10.1137\/07068237x","relation":{},"ISSN":["0036-1429","1095-7170"],"issn-type":[{"value":"0036-1429","type":"print"},{"value":"1095-7170","type":"electronic"}],"subject":[],"published":{"date-parts":[[2008,1]]}}}