{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,7,1]],"date-time":"2025-07-01T13:48:55Z","timestamp":1751377735963},"reference-count":26,"publisher":"Springer Science and Business Media LLC","issue":"2","license":[{"start":{"date-parts":[[1993,6,1]],"date-time":"1993-06-01T00:00:00Z","timestamp":738892800000},"content-version":"tdm","delay-in-days":0,"URL":"http:\/\/www.springer.com\/tdm"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Adv Comput Math"],"published-print":{"date-parts":[[1993,6]]},"DOI":"10.1007\/bf02071387","type":"journal-article","created":{"date-parts":[[2005,8,13]],"date-time":"2005-08-13T11:53:36Z","timestamp":1123934016000},"page":"223-239","source":"Crossref","is-referenced-by-count":10,"title":["Error bounds for the finite element approximation of a degenerate quasilinear parabolic variational inequality"],"prefix":"10.1007","volume":"1","author":[{"given":"W. B.","family":"Liu","sequence":"first","affiliation":[]},{"given":"John W.","family":"Barrett","sequence":"additional","affiliation":[]}],"member":"297","reference":[{"key":"BF02071387_CR1","doi-asserted-by":"crossref","first-page":"495","DOI":"10.1017\/S030500410005636X","volume":"86","author":"C. Atkinson","year":"1979","unstructured":"C. Atkinson and J.E. Bouillet, Some qualitative properties of solutions of a generalised diffusion equation, Math. Proc. Camb. Phil. Soc. 86(1979)495\u2013510.","journal-title":"Math. Proc. Camb. Phil. Soc."},{"key":"BF02071387_CR2","doi-asserted-by":"crossref","first-page":"401","DOI":"10.1093\/qjmam\/37.3.401","volume":"37","author":"C. Atkinson","year":"1984","unstructured":"C. Atkinson and C.R. Champion, Some boundary-value problems for the equation \u2207 \u00b7 (|\u2207\u03d5| N \u2207\u03d5)=0, Q. J. Mech. Appl. Math. 37(1984)401\u2013419.","journal-title":"Q. J. Mech. Appl. Math."},{"key":"BF02071387_CR3","doi-asserted-by":"crossref","first-page":"193","DOI":"10.1093\/qjmam\/27.2.193","volume":"27","author":"C. Atkinson","year":"1974","unstructured":"C. Atkinson and C.W. Jones, Similarity solutions in some non-linear diffusion problems and in boundary-layer flow of a pseudo plastic fluid, Q. J. Mech. Appl. Math. 27(1974)193\u2013211.","journal-title":"Q. J. Mech. Appl. Math."},{"key":"BF02071387_CR4","doi-asserted-by":"crossref","DOI":"10.1007\/978-94-010-1537-0","volume-title":"Nonlinear Semigroups and Differential Equations in Banach Spaces","author":"V. Barbu","year":"1976","unstructured":"V. Barbu,Nonlinear Semigroups and Differential Equations in Banach Spaces (Noordhoff, Leiden, 1976)."},{"key":"BF02071387_CR5","unstructured":"J.W. Barrett and W.B. Liu, Finite element approximation of thep-Laplacian, Math. Comp., to appear."},{"key":"BF02071387_CR6","unstructured":"J.W. Barrett and W.B. Liu, Finite element approximation of the parabolicp-Laplacian, submitted for publication."},{"key":"BF02071387_CR7","first-page":"1","volume":"51","author":"H. Br\u00e9zis","year":"1972","unstructured":"H. Br\u00e9zis, Probl\u00e8mes unilat\u00e9raux, J. Math. Pure Appl. 51(1972)1\u2013164.","journal-title":"J. Math. Pure Appl."},{"key":"BF02071387_CR8","doi-asserted-by":"crossref","first-page":"383","DOI":"10.1007\/BF00376141","volume":"114","author":"H.J. Choe","year":"1991","unstructured":"H.J. Choe, A regularity theory for a general class of quasilinear elliptic partial differential equations and obstacle problems, Arch. Rat. Mech. Anal. 114(1991)383\u2013394.","journal-title":"Arch. Rat. Mech. Anal."},{"key":"BF02071387_CR9","unstructured":"H.J. Choe, A regularity theory for a more general class of quasilinear parabolic partial differential equations and variational inequalities, Report 756, Institute for Mathematics and its Applications, University of Minnesota (1990)."},{"key":"BF02071387_CR10","doi-asserted-by":"crossref","first-page":"623","DOI":"10.1137\/0522039","volume":"22","author":"H.J. Choe","year":"1991","unstructured":"H.J. Choe and J.L. Lewis, On the obstacle problem for quasilinear elliptic equations ofp-Laplacian type, SIAM J. Math. Anal. 22(1991)623\u2013638.","journal-title":"SIAM J. Math. Anal."},{"key":"BF02071387_CR11","doi-asserted-by":"crossref","first-page":"373","DOI":"10.1007\/BF01396320","volume":"54","author":"S.S. Chow","year":"1988","unstructured":"S.S. Chow, Finite element error estimates for non-linear elliptic equations of monotone type, Numer. Math. 54(1988)373\u2013393.","journal-title":"Numer. Math."},{"key":"BF02071387_CR12","doi-asserted-by":"crossref","first-page":"51","DOI":"10.1007\/BF02165273","volume":"13","author":"P.G. Ciarlet","year":"1969","unstructured":"P.G. Ciarlet, M.H. Schultz and R.S. Varga, Numerical methods of high order accuracy for nonlinear boundary value problems, V. Montone operator theory, Numer. Math. 13(1969)51\u201377.","journal-title":"Numer. Math."},{"key":"BF02071387_CR13","volume-title":"The Finite Element Method for Elliptic Problems","author":"P.G. Ciarlet","year":"1978","unstructured":"P.G. Ciarlet,The Finite Element Method for Elliptic Problems (North-Holland, Amsterdam, 1978)."},{"key":"BF02071387_CR14","doi-asserted-by":"crossref","first-page":"155","DOI":"10.1002\/mana.19800940112","volume":"94","author":"M. Dobrowolski","year":"1980","unstructured":"M. Dobrowolski and R. Rannacher, Finite element methods for nonlinear elliptic systems of second order, Math. Nachr. 94(1980)155\u2013172.","journal-title":"Math. Nachr."},{"key":"BF02071387_CR15","doi-asserted-by":"crossref","first-page":"963","DOI":"10.1090\/S0025-5718-1974-0391502-8","volume":"28","author":"R.S. Falk","year":"1974","unstructured":"R.S. Falk, Error estimates for the approximation of a class of variational inequalities, Math. Comp. 28(1974)963\u2013971.","journal-title":"Math. Comp."},{"key":"BF02071387_CR16","doi-asserted-by":"crossref","first-page":"557","DOI":"10.1007\/BF01408576","volume":"50","author":"A. Fetter","year":"1987","unstructured":"A. Fetter,L \u221e-error estimate for an approximation of a parabolic variational inequality, Numer. Math. 50(1987)557\u2013565.","journal-title":"Numer. Math."},{"key":"BF02071387_CR17","doi-asserted-by":"crossref","first-page":"419","DOI":"10.1137\/0715026","volume":"15","author":"J. Frehse","year":"1978","unstructured":"J. Frehse and R. Rannacher, AsymptoticL \u221e-error estimates for linear finite element approximations of quasilinear boundary problems, SIAM J. Numer. Anal. 15(1978)419\u2013431.","journal-title":"SIAM J. Numer. Anal."},{"key":"BF02071387_CR18","doi-asserted-by":"crossref","first-page":"41","DOI":"10.1051\/m2an\/197509R200411","volume":"2","author":"R. Glowinski","year":"1975","unstructured":"R. Glowinski and A. Marrocco, Sur l'approximation par \u00e9l\u00e9ments finis d'ordre un, et la r\u00e9solution, par p\u00e9nalisation-dualit\u00e9, d'une classe de probl\u00e8mes de Dirichlet non lin\u00e9aires, R.A.I.R.O. Analyse Num\u00e9rique 2(1975)41\u201364.","journal-title":"R.A.I.R.O. Analyse Num\u00e9rique"},{"key":"BF02071387_CR19","doi-asserted-by":"crossref","first-page":"599","DOI":"10.1137\/0713050","volume":"13","author":"C. Johnson","year":"1976","unstructured":"C. Johnson, A convergence estimate for an approximation of a parabolic variational inequality, SIAM J. Numer. Anal. 13(1976)599\u2013607.","journal-title":"SIAM J. Numer. Anal."},{"key":"BF02071387_CR20","unstructured":"W.B. Liu and J.W. Barrett, Finite element approximation of some degenerate monotone quasilinear elliptic systems, submitted for publication."},{"key":"BF02071387_CR21","unstructured":"W.B. Liu and J.W. Barrett, A further remark on the regularity of the solutions of thep-Laplacian and its applications to their finite element approximation, Nonlinear Anal., to appear."},{"key":"BF02071387_CR22","unstructured":"W.B. Liu and J.W. Barrett, Quasi-norm error bounds for the finite element approximation of some degenerate quasilinear elliptic equations and variational inequalities, submitted for publication."},{"key":"BF02071387_CR23","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1071\/PH610001","volume":"14","author":"J.R. Philip","year":"1961","unstructured":"J.R. Philip,N-diffusion, Aust. J. Phys. 14(1961)1\u201313.","journal-title":"Aust. J. Phys."},{"key":"BF02071387_CR24","doi-asserted-by":"crossref","first-page":"453","DOI":"10.1007\/BF01386423","volume":"57","author":"C. Vuik","year":"1990","unstructured":"C. Vuik, AnL 2-error estimate for an approximation of the solution of a parabolic variational inequality, Numer. Math. 57(1990)453\u2013471.","journal-title":"Numer. Math."},{"key":"BF02071387_CR25","volume-title":"Variational Inequalities","author":"Y.D. Wang","year":"1987","unstructured":"Y.D. Wang,Variational Inequalities (Higher Education Press, Beijing, 1987), in Chinese."},{"key":"BF02071387_CR26","doi-asserted-by":"crossref","first-page":"484","DOI":"10.1137\/0729029","volume":"29","author":"D. Wei","year":"1992","unstructured":"D. Wei, Existence, uniqueness and numerical analysis of solutions of a quasilinear parabolic problem, SIAM J. Numer. Anal. 29(1992)484\u2013497.","journal-title":"SIAM J. Numer. Anal."}],"container-title":["Advances in Computational Mathematics"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/BF02071387.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/article\/10.1007\/BF02071387\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/BF02071387","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,5,14]],"date-time":"2019-05-14T00:06:34Z","timestamp":1557792394000},"score":1,"resource":{"primary":{"URL":"http:\/\/link.springer.com\/10.1007\/BF02071387"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1993,6]]},"references-count":26,"journal-issue":{"issue":"2","published-print":{"date-parts":[[1993,6]]}},"alternative-id":["BF02071387"],"URL":"https:\/\/doi.org\/10.1007\/bf02071387","relation":{},"ISSN":["1019-7168","1572-9044"],"issn-type":[{"value":"1019-7168","type":"print"},{"value":"1572-9044","type":"electronic"}],"subject":[],"published":{"date-parts":[[1993,6]]}}}