{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,7]],"date-time":"2025-10-07T12:11:12Z","timestamp":1759839072357},"reference-count":40,"publisher":"Springer Science and Business Media LLC","issue":"5","license":[{"start":{"date-parts":[[2014,2,19]],"date-time":"2014-02-19T00:00:00Z","timestamp":1392768000000},"content-version":"tdm","delay-in-days":0,"URL":"http:\/\/www.springer.com\/tdm"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Found Comput Math"],"published-print":{"date-parts":[[2014,10]]},"DOI":"10.1007\/s10208-013-9183-7","type":"journal-article","created":{"date-parts":[[2014,2,18]],"date-time":"2014-02-18T20:00:55Z","timestamp":1392753655000},"page":"863-912","source":"Crossref","is-referenced-by-count":7,"title":["Convergence Analysis of Spatially Adaptive Rothe Methods"],"prefix":"10.1007","volume":"14","author":[{"given":"Petru A.","family":"Cioica","sequence":"first","affiliation":[]},{"given":"Stephan","family":"Dahlke","sequence":"additional","affiliation":[]},{"given":"Nicolas","family":"D\u00f6hring","sequence":"additional","affiliation":[]},{"given":"Ulrich","family":"Friedrich","sequence":"additional","affiliation":[]},{"given":"Stefan","family":"Kinzel","sequence":"additional","affiliation":[]},{"given":"Felix","family":"Lindner","sequence":"additional","affiliation":[]},{"given":"Thorsten","family":"Raasch","sequence":"additional","affiliation":[]},{"given":"Klaus","family":"Ritter","sequence":"additional","affiliation":[]},{"given":"Ren\u00e9 L.","family":"Schilling","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2014,2,19]]},"reference":[{"key":"9183_CR1","doi-asserted-by":"crossref","unstructured":"I. Babu\u0161ka, Advances in the p and h-p versions of the finite element method. A survey, Numerical mathematics, Proc. Int. Conf., Singapore 1988, ISNM, Int. Ser. Numer. Math. 86 (1988), 31\u201346.","DOI":"10.1007\/978-3-0348-6303-2_3"},{"key":"9183_CR2","unstructured":"I. Babu\u0161ka and W.C. Rheinboldt, A survey of a posteriori error estimators and adaptive approaches in the finite element method, Finite element methods. Proc. China-France Symp., Beijing\/China (1983), 1\u201356."},{"key":"9183_CR3","doi-asserted-by":"crossref","unstructured":"R.E. Bank and A. Weiser, Some a posteriori error estimators for elliptic partial differential equations, Math. Comput. 44 (1985), 283\u2013301.","DOI":"10.1090\/S0025-5718-1985-0777265-X"},{"key":"9183_CR4","doi-asserted-by":"crossref","unstructured":"J. Bergh and J. L\u00f6fstr\u00f6m, Interpolation Spaces. An Introduction, Springer, Berlin, 1976.","DOI":"10.1007\/978-3-642-66451-9"},{"key":"9183_CR5","doi-asserted-by":"crossref","unstructured":"P. Binev, W. Dahmen, and R. DeVore, Adaptive finite element methods with convergence rates, Numer. Math. 97 (2004), 219\u2013268.","DOI":"10.1007\/s00211-003-0492-7"},{"key":"9183_CR6","doi-asserted-by":"crossref","unstructured":"F.A. Bornemann, B. Erdmann, and R. Kornhuber, A posteriori error estimates for elliptic problems in two and three space dimensions, SIAM J. Numer. Anal. 33 (1996), 1188\u20131204.","DOI":"10.1137\/0733059"},{"key":"9183_CR7","doi-asserted-by":"crossref","unstructured":"C. Canuto, A. Tabacco, and K. Urban, The wavelet element method. II: Realization and additional features in 2D and 3D, Appl. Comput. Harmon. Anal. 8 (2000), 123\u2013165.","DOI":"10.1006\/acha.2000.0282"},{"key":"9183_CR8","doi-asserted-by":"crossref","unstructured":"A. Cohen, Wavelet Methods in Numerical Analysis, North-Holland\/Elsevier, Amsterdam, 2000.","DOI":"10.1016\/S1570-8659(00)07004-6"},{"key":"9183_CR9","doi-asserted-by":"crossref","unstructured":"A. Cohen, W. Dahmen, and R.A. DeVore, Adaptive wavelet methods for elliptic operator equations: Convergence rates, Math. Comput. 70 (2001), 27\u201375.","DOI":"10.1090\/S0025-5718-00-01252-7"},{"key":"9183_CR10","doi-asserted-by":"crossref","unstructured":"A. Cohen, W. Dahmen, and R.A. DeVore, Adaptive wavelet methods. II: Beyond the elliptic case, Found. Comput. Math. 2 (2002), 203\u2013245.","DOI":"10.1007\/s102080010027"},{"key":"9183_CR11","doi-asserted-by":"crossref","unstructured":"M. Crouzeix and V. Thom\u00e9e, On the discretization in time of semilinear parabolic equations with nonsmooth initial data, Math. Comput. 49 (1987), 359\u2013377.","DOI":"10.1090\/S0025-5718-1987-0906176-3"},{"key":"9183_CR12","doi-asserted-by":"crossref","unstructured":"S. Dahlke, Besov regularity for elliptic boundary value problems in polygonal domains, Appl. Math. Lett. 12 (1999), 31\u201336.","DOI":"10.1016\/S0893-9659(99)00075-0"},{"key":"9183_CR13","doi-asserted-by":"crossref","unstructured":"S. Dahlke, W. Dahmen, and R.A. DeVore, Nonlinear approximation and adaptive techniques for solving elliptic operator equations, Multiscale wavelet methods for partial differential equations (W. Dahmen, A. Kurdila, and P. Oswald, eds.), Academic Press, San Diego, 1997, pp. 237\u2013284.","DOI":"10.1016\/S1874-608X(97)80008-8"},{"key":"9183_CR14","doi-asserted-by":"crossref","unstructured":"S. Dahlke, W. Dahmen, R. Hochmuth, and R. Schneider, Stable multiscale bases and local error estimation for elliptic problems, Appl. Numer. Math. 23 (1997), 21\u201347.","DOI":"10.1016\/S0168-9274(96)00060-8"},{"key":"9183_CR15","doi-asserted-by":"crossref","unstructured":"S. Dahlke and R.A. DeVore, Besov regularity for elliptic boundary value problems, Commun. Partial Differ. Equations 22 (1997), 1\u201316.","DOI":"10.1080\/03605309708821252"},{"key":"9183_CR16","doi-asserted-by":"crossref","unstructured":"S. Dahlke, M. Fornasier, T. Raasch, R. Stevenson, and M. Werner, Adaptive frame methods for elliptic operator equations: The steepest descent approach, IMA J. Numer. Anal. 27 (2007), 717\u2013740.","DOI":"10.1093\/imanum\/drl035"},{"key":"9183_CR17","doi-asserted-by":"crossref","unstructured":"S. Dahlke, E. Novak, and W. Sickel, Optimal approximation of elliptic problems by linear and nonlinear mappings. I, J. Complexity 22 (2006), 29\u201349.","DOI":"10.1016\/j.jco.2005.06.005"},{"key":"9183_CR18","doi-asserted-by":"crossref","unstructured":"S. Dahlke and W. Sickel, On Besov regularity of solutions to nonlinear elliptic partial differential equations, Rev. Mat. Complut. 26 (2013), 115\u2013145.","DOI":"10.1007\/s13163-012-0093-z"},{"key":"9183_CR19","doi-asserted-by":"crossref","unstructured":"W. Dahmen and R. Schneider, Wavelets with complementary boundary conditions \u2013 functions spaces on the cube, Result. Math. 34 (1998), 255\u2013293.","DOI":"10.1007\/BF03322055"},{"key":"9183_CR20","doi-asserted-by":"crossref","unstructured":"W. Dahmen and R. Schneider, Composite wavelet bases for operator equations, Math. Comput. 68 (1999), 1533\u20131567.","DOI":"10.1090\/S0025-5718-99-01092-3"},{"key":"9183_CR21","doi-asserted-by":"crossref","unstructured":"W. Dahmen and R. Schneider, Wavelets on manifolds. I: Construction and domain decomposition, SIAM J. Math. Anal. 31 (1999), 184\u2013230.","DOI":"10.1137\/S0036141098333451"},{"key":"9183_CR22","doi-asserted-by":"crossref","unstructured":"R.A. DeVore, Nonlinear approximation, Acta Numerica 7 (1998), 51\u2013150.","DOI":"10.1017\/S0962492900002816"},{"key":"9183_CR23","doi-asserted-by":"crossref","unstructured":"W. D\u00f6rfler, A convergent adaptive algorithm for Poisson\u2019s equation, SIAM J. Numer. Anal. 33 (1996), 737\u2013785.","DOI":"10.1137\/0733054"},{"key":"9183_CR24","doi-asserted-by":"crossref","unstructured":"K. Eriksson, An adaptive finite element method with efficient maximum norm error control for elliptic problems, Math. Models Methods Appl. Sci. 4 (1994), 313\u2013329.","DOI":"10.1142\/S0218202594000194"},{"key":"9183_CR25","doi-asserted-by":"crossref","unstructured":"K. Eriksson and C. Johnson, Adaptive finite element methods for parabolic problems. I: A linear model problem, SIAM J. Numer. Anal. 28 (1991), 43\u201377.","DOI":"10.1137\/0728003"},{"key":"9183_CR26","doi-asserted-by":"crossref","unstructured":"K. Eriksson and C. Johnson, Adaptive finite element methods for parabolic problems. II: Optimal error estimates in $$L_\\infty L_2$$ L \u221e L 2 and $$L_\\infty L_\\infty $$ L \u221e L \u221e , SIAM J. Numer. Anal. 32 (1995), 706\u2013740.","DOI":"10.1137\/0732033"},{"key":"9183_CR27","doi-asserted-by":"crossref","unstructured":"K. Eriksson, C. Johnson, and S. Larsson, Adaptive finite element methods for parabolic problems. VI: Analytic semigroups, SIAM J. Numer. Anal. 35 (1998), 1315\u20131325.","DOI":"10.1137\/S0036142996310216"},{"key":"9183_CR28","unstructured":"M. Hanke-Bourgeois, Foundations of Numerical Mathematics and Scientific Computing, Vieweg+Teubner, Wiesbaden, 2009."},{"key":"9183_CR29","doi-asserted-by":"crossref","unstructured":"P. Hansbo and C. Johnson, Adaptive finite element methods in computational mechanics, Comput. Methods Appl. Mech. Eng. 101 (1992), 143\u2013181.","DOI":"10.1016\/0045-7825(92)90020-K"},{"key":"9183_CR30","doi-asserted-by":"crossref","unstructured":"D. Jerison and C.E. Kenig, The inhomogeneous Dirichlet problem in Lipschitz domains, J. Funct. Anal. 130 (1995), 161\u2013219.","DOI":"10.1006\/jfan.1995.1067"},{"key":"9183_CR31","unstructured":"C. Johnson, Numerical Solution of Partial Differential Equations by the Finite Element Method, Dover Publications, Mineola, 2009."},{"key":"9183_CR32","unstructured":"T. Kato, Perturbation Theory for Linear Operators. 2nd corr. print. of the 2nd ed., Springer, Berlin, 1984."},{"key":"9183_CR33","doi-asserted-by":"crossref","unstructured":"J. Lang, Adaptive Multilevel Solution of Nonlinear Parabolic PDE Systems. Theory, Algorithm, and Applications, Springer, Berlin, 2001.","DOI":"10.1007\/978-3-662-04484-1"},{"key":"9183_CR34","doi-asserted-by":"crossref","unstructured":"C. Lubich and A. Ostermann, Linearly implicit time discretization of nonlinear parabolic equations, IMA J. Numer. Anal. 15 (1995), 555\u2013583.","DOI":"10.1093\/imanum\/15.4.555"},{"key":"9183_CR35","doi-asserted-by":"crossref","unstructured":"A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Applied Mathematical Sciences, 44. New York: Springer-Verlag, 1983.","DOI":"10.1007\/978-1-4612-5561-1"},{"key":"9183_CR36","doi-asserted-by":"crossref","unstructured":"R. Stevenson, Optimality of a standard adaptive finite element method, Found. Comput. Math. 7 (2007), 245\u2013269.","DOI":"10.1007\/s10208-005-0183-0"},{"key":"9183_CR37","unstructured":"V. Thom\u00e9e, Galerkin Finite Element Methods for Parabolic Problems, Springer, Berlin, 2006."},{"key":"9183_CR38","doi-asserted-by":"crossref","unstructured":"R. Verf\u00fcrth, A posteriori error estimation and adaptive mesh-refinement techniques, J. Comput. Appl. Math. 50 (1994), 67\u201383.","DOI":"10.1016\/0377-0427(94)90290-9"},{"key":"9183_CR39","unstructured":"R. Verf\u00fcrth, A Review of A Posteriori Error Estimation and Adaptive Mesh-Refinement Techniques, Wiley-Teubner Series Advances in Numerical Mathematics. Chichester: Wiley. Stuttgart: B. G. Teubner, 1996."},{"key":"9183_CR40","doi-asserted-by":"crossref","unstructured":"J.G. Verwer, E.J. Spee, J.G. Blom, and W. Hundsdorfer, A second-order Rosenbrock method applied to photochemical dispersion problems, SIAM J. Sci. Comput. 20 (1999), 1456\u20131480.","DOI":"10.1137\/S1064827597326651"}],"container-title":["Foundations of Computational Mathematics"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s10208-013-9183-7.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/article\/10.1007\/s10208-013-9183-7\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s10208-013-9183-7","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,8,7]],"date-time":"2019-08-07T19:02:32Z","timestamp":1565204552000},"score":1,"resource":{"primary":{"URL":"http:\/\/link.springer.com\/10.1007\/s10208-013-9183-7"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2014,2,19]]},"references-count":40,"journal-issue":{"issue":"5","published-print":{"date-parts":[[2014,10]]}},"alternative-id":["9183"],"URL":"https:\/\/doi.org\/10.1007\/s10208-013-9183-7","relation":{},"ISSN":["1615-3375","1615-3383"],"issn-type":[{"value":"1615-3375","type":"print"},{"value":"1615-3383","type":"electronic"}],"subject":[],"published":{"date-parts":[[2014,2,19]]}}}