{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,1]],"date-time":"2026-03-01T10:29:05Z","timestamp":1772360945014,"version":"3.50.1"},"reference-count":11,"publisher":"Walter de Gruyter GmbH","issue":"2","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"published-print":{"date-parts":[[2018,4,1]]},"abstract":"Abstract<\/jats:title>\n The stochastic Allen\u2013Cahn equation with multiplicative noise involves\nthe nonlinear drift operator \n \n \n \n \n \ud835\udc9c<\/m:mi>\n \u2062<\/m:mo>\n \n (<\/m:mo>\n x<\/m:mi>\n )<\/m:mo>\n <\/m:mrow>\n <\/m:mrow>\n =<\/m:mo>\n \n \n \u0394<\/m:mi>\n \u2062<\/m:mo>\n x<\/m:mi>\n <\/m:mrow>\n -<\/m:mo>\n \n \n (<\/m:mo>\n \n \n \n |<\/m:mo>\n x<\/m:mi>\n |<\/m:mo>\n <\/m:mrow>\n 2<\/m:mn>\n <\/m:msup>\n -<\/m:mo>\n 1<\/m:mn>\n <\/m:mrow>\n )<\/m:mo>\n <\/m:mrow>\n \u2062<\/m:mo>\n x<\/m:mi>\n <\/m:mrow>\n <\/m:mrow>\n <\/m:mrow>\n <\/m:math>\n \n {{\\mathscr{A}}(x)=\\Delta x-(|x|^{2}-1)x}<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula>. We use the fact that \n \n \n \n \n \ud835\udc9c<\/m:mi>\n \u2062<\/m:mo>\n \n (<\/m:mo>\n x<\/m:mi>\n )<\/m:mo>\n <\/m:mrow>\n <\/m:mrow>\n =<\/m:mo>\n \n -<\/m:mo>\n \n \n \ud835\udca5<\/m:mi>\n \u2032<\/m:mo>\n <\/m:msup>\n \u2062<\/m:mo>\n \n (<\/m:mo>\n x<\/m:mi>\n )<\/m:mo>\n <\/m:mrow>\n <\/m:mrow>\n <\/m:mrow>\n <\/m:mrow>\n <\/m:math>\n \n {{\\mathscr{A}}(x)=-{\\mathcal{J}}^{\\prime}(x)}<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\nsatisfies a weak monotonicity property to deduce uniform bounds\nin strong norms for solutions of the temporal, as well as of the spatio-temporal discretization of the problem. This weak monotonicity property then allows\nfor the estimate<\/jats:p>\n \n \n \n \n \n \n \n sup<\/m:mo>\n \n 1<\/m:mn>\n \u2264<\/m:mo>\n j<\/m:mi>\n \u2264<\/m:mo>\n J<\/m:mi>\n <\/m:mrow>\n <\/m:munder>\n \u2061<\/m:mo>\n \n \ud835\udd3c<\/m:mi>\n \u2062<\/m:mo>\n \n [<\/m:mo>\n \n \n \u2225<\/m:mo>\n \n \n X<\/m:mi>\n \n t<\/m:mi>\n j<\/m:mi>\n <\/m:msub>\n <\/m:msub>\n -<\/m:mo>\n \n Y<\/m:mi>\n j<\/m:mi>\n <\/m:msup>\n <\/m:mrow>\n \u2225<\/m:mo>\n <\/m:mrow>\n \n \ud835\udd43<\/m:mi>\n 2<\/m:mn>\n <\/m:msup>\n 2<\/m:mn>\n <\/m:msubsup>\n ]<\/m:mo>\n <\/m:mrow>\n <\/m:mrow>\n <\/m:mrow>\n \u2264<\/m:mo>\n \n \n C<\/m:mi>\n \u03b4<\/m:mi>\n <\/m:msub>\n \u2062<\/m:mo>\n \n (<\/m:mo>\n \n \n k<\/m:mi>\n \n 1<\/m:mn>\n -<\/m:mo>\n \u03b4<\/m:mi>\n <\/m:mrow>\n <\/m:msup>\n +<\/m:mo>\n \n h<\/m:mi>\n 2<\/m:mn>\n <\/m:msup>\n <\/m:mrow>\n )<\/m:mo>\n <\/m:mrow>\n <\/m:mrow>\n <\/m:mrow>\n <\/m:math>\n \n \\sup_{1\\leq j\\leq J}{\\mathbb{E}}[\\|X_{t_{j}}-Y^{j}\\|_{{\\mathbb{L}}^{2}}^{2}]%\n\\leq C_{\\delta}(k^{1-\\delta}+h^{2})<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:disp-formula>\n <\/jats:p>\n for all small \n \n \n \n \u03b4<\/m:mi>\n ><\/m:mo>\n 0<\/m:mn>\n <\/m:mrow>\n <\/m:math>\n \n {\\delta>0}<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula>, where\nX<\/jats:italic> is the strong variational solution of the stochastic Allen\u2013Cahn equation, while \n \n \n \n {<\/m:mo>\n \n Y<\/m:mi>\n j<\/m:mi>\n <\/m:msup>\n :<\/m:mo>\n \n 0<\/m:mn>\n \u2264<\/m:mo>\n j<\/m:mi>\n \u2264<\/m:mo>\n J<\/m:mi>\n <\/m:mrow>\n }<\/m:mo>\n <\/m:mrow>\n <\/m:math>\n \n {\\{Y^{j}:0\\leq j\\leq J\\}}<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula> solves a structure preserving finite element based space-time discretization\nof the problem on a temporal mesh \n \n \n \n {<\/m:mo>\n \n t<\/m:mi>\n j<\/m:mi>\n <\/m:msub>\n :<\/m:mo>\n \n 1<\/m:mn>\n \u2264<\/m:mo>\n j<\/m:mi>\n \u2264<\/m:mo>\n J<\/m:mi>\n <\/m:mrow>\n }<\/m:mo>\n <\/m:mrow>\n <\/m:math>\n \n {\\{t_{j}:1\\leq j\\leq J\\}}<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula> of size \n \n \n \n k<\/m:mi>\n ><\/m:mo>\n 0<\/m:mn>\n <\/m:mrow>\n <\/m:math>\n \n {k>0}<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula> which covers \n \n \n \n [<\/m:mo>\n 0<\/m:mn>\n ,<\/m:mo>\n T<\/m:mi>\n ]<\/m:mo>\n <\/m:mrow>\n <\/m:math>\n \n {[0,T]}<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula>.<\/jats:p>","DOI":"10.1515\/cmam-2017-0023","type":"journal-article","created":{"date-parts":[[2017,7,18]],"date-time":"2017-07-18T10:00:56Z","timestamp":1500372056000},"page":"297-311","source":"Crossref","is-referenced-by-count":38,"title":["Optimal Strong Rates of Convergence for a Space-Time Discretization of\nthe Stochastic Allen\u2013Cahn Equation with Multiplicative Noise"],"prefix":"10.1515","volume":"18","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-1507-5772","authenticated-orcid":false,"given":"Ananta K.","family":"Majee","sequence":"first","affiliation":[{"name":"Mathematisches Institut , Universit\u00e4t T\u00fcbingen , Auf der Morgenstelle 10, 72076 T\u00fcbingen , Germany"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Andreas","family":"Prohl","sequence":"additional","affiliation":[{"name":"Mathematisches Institut , Universit\u00e4t T\u00fcbingen , Auf der Morgenstelle 10, 72076 T\u00fcbingen , Germany"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"374","published-online":{"date-parts":[[2017,7,18]]},"reference":[{"key":"2023033109580940192_j_cmam-2017-0023_ref_001_w2aab3b7e2183b1b6b1ab2ab1Aa","doi-asserted-by":"crossref","unstructured":"S. C. Brenner and L. R. Scott,\nThe Mathematical Theory of Finite Element Methods, 2nd ed.,\nTexts Appl. Math. 15,\nSpringer, New York, 2002.","DOI":"10.1007\/978-1-4757-3658-8"},{"key":"2023033109580940192_j_cmam-2017-0023_ref_002_w2aab3b7e2183b1b6b1ab2ab2Aa","doi-asserted-by":"crossref","unstructured":"E. Carelli and A. Prohl,\nRates of convergence for discretizations of the stochastic incompressible Navier\u2013Stokes equations,\nSIAM J. Numer. Anal. 50 (2012), no. 5, 2467\u20132496.","DOI":"10.1137\/110845008"},{"key":"2023033109580940192_j_cmam-2017-0023_ref_003_w2aab3b7e2183b1b6b1ab2ab3Aa","doi-asserted-by":"crossref","unstructured":"J. Douglas, Jr., T. Dupont and L. Wahlbin,\nThe stability in LqL^{q} of the L2L^{2}-projection into finite element function spaces,\nNumer. Math. 23 (1974\/75), 193\u2013197.","DOI":"10.1007\/BF01400302"},{"key":"2023033109580940192_j_cmam-2017-0023_ref_004_w2aab3b7e2183b1b6b1ab2ab4Aa","doi-asserted-by":"crossref","unstructured":"B. Gess,\nStrong solutions for stochastic partial differential equations of gradient type,\nJ. Funct. Anal. 263 (2012), no. 8, 2355\u20132383.","DOI":"10.1016\/j.jfa.2012.07.001"},{"key":"2023033109580940192_j_cmam-2017-0023_ref_005_w2aab3b7e2183b1b6b1ab2ab5Aa","doi-asserted-by":"crossref","unstructured":"I. Gy\u00f6ngy and A. Millet,\nRate of convergence of implicit approximations for stochastic evolution equations,\nStochastic Differential Equations: Theory and Applications,\nInterdiscip. Math. Sci. 2,\nWorld Scientific, Hackensack (2007), 281\u2013310.","DOI":"10.1142\/9789812770639_0011"},{"key":"2023033109580940192_j_cmam-2017-0023_ref_006_w2aab3b7e2183b1b6b1ab2ab6Aa","doi-asserted-by":"crossref","unstructured":"I. Gy\u00f6ngy and A. Millet,\nRate of convergence of space time approximations for stochastic evolution equations,\nPotential Anal. 30 (2009), no. 1, 29\u201364.","DOI":"10.1007\/s11118-008-9105-5"},{"key":"2023033109580940192_j_cmam-2017-0023_ref_007_w2aab3b7e2183b1b6b1ab2ab7Aa","doi-asserted-by":"crossref","unstructured":"D. J. Higham, X. Mao and A. M. Stuart,\nStrong convergence of Euler-type methods for nonlinear stochastic differential equations,\nSIAM J. Numer. Anal. 40 (2002), no. 3, 1041\u20131063.","DOI":"10.1137\/S0036142901389530"},{"key":"2023033109580940192_j_cmam-2017-0023_ref_008_w2aab3b7e2183b1b6b1ab2ab8Aa","doi-asserted-by":"crossref","unstructured":"M. Kov\u00e1cs, S. Larsson and A. Mesforush,\nFinite element approximation of the Cahn\u2013Hilliard\u2013Cook equation,\nSIAM J. Numer. Anal. 49 (2011), no. 6, 2407\u20132429.","DOI":"10.1137\/110828150"},{"key":"2023033109580940192_j_cmam-2017-0023_ref_009_w2aab3b7e2183b1b6b1ab2ab9Aa","doi-asserted-by":"crossref","unstructured":"J. Printems,\nOn the discretization in time of parabolic stochastic partial differential equations,\nESAIM Math. Model. Numer. Anal. 35 (2001), no. 6, 1055\u20131078.","DOI":"10.1051\/m2an:2001148"},{"key":"2023033109580940192_j_cmam-2017-0023_ref_010_w2aab3b7e2183b1b6b1ab2ac10Aa","unstructured":"A. Prohl,\nStrong rates of convergence for a space-time discretization of the stochastic Allen\u2013Cahn equation with multiplicative noise,\npreprint (2014), https:\/\/na.uni-tuebingen.de\/preprints."},{"key":"2023033109580940192_j_cmam-2017-0023_ref_011_w2aab3b7e2183b1b6b1ab2ac11Aa","doi-asserted-by":"crossref","unstructured":"M. Sauer and W. Stannat,\nLattice approximation for stochastic reaction diffusion equations with one-sided Lipschitz condition,\nMath. 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