{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,11,5]],"date-time":"2025-11-05T13:38:43Z","timestamp":1762349923928,"version":"build-2065373602"},"reference-count":34,"publisher":"Walter de Gruyter GmbH","issue":"4","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"published-print":{"date-parts":[[2025,10,1]]},"abstract":"<jats:title>Abstract<\/jats:title>\n                  <jats:p>\n                    In this contribution, we provide convergence rates for a finite volume scheme of the stochastic heat equation with multiplicative Lipschitz noise and homogeneous Neumann boundary conditions (SHE). More precisely, we give an error estimate for the\n                    <jats:inline-formula id=\"j_cmam-2024-0089_ineq_9999\">\n                      <jats:alternatives>\n                        <m:math xmlns:m=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                          <m:msup>\n                            <m:mi>L<\/m:mi>\n                            <m:mn>2<\/m:mn>\n                          <\/m:msup>\n                        <\/m:math>\n                        <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" xlink:href=\"graphic\/j_cmam-2024-0089_eq_0405.png\"\/>\n                        <jats:tex-math>{L^{2}}<\/jats:tex-math>\n                      <\/jats:alternatives>\n                    <\/jats:inline-formula>\n                    -norm of the space-time discretization of SHE by a semi-implicit Euler scheme with respect to time and a TPFA scheme with respect to space and the variational solution of SHE. The only regularity assumptions additionally needed is spatial regularity of the initial datum and smoothness of the diffusive term.\n                  <\/jats:p>","DOI":"10.1515\/cmam-2024-0089","type":"journal-article","created":{"date-parts":[[2025,7,8]],"date-time":"2025-07-08T06:41:06Z","timestamp":1751956866000},"page":"981-1002","source":"Crossref","is-referenced-by-count":0,"title":["Convergence Rates for a Finite Volume Scheme of the Stochastic Heat Equation"],"prefix":"10.1515","volume":"25","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-8740-7086","authenticated-orcid":false,"given":"Niklas","family":"Sapountzoglou","sequence":"first","affiliation":[{"name":"Institute of Mathematics , 160488 Clausthal University of Technology , 38678 Clausthal-Zellerfeld , Germany"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-5792-6803","authenticated-orcid":false,"given":"Aleksandra","family":"Zimmermann","sequence":"additional","affiliation":[{"name":"Institute of Mathematics , 160488 Clausthal University of Technology , 38678 Clausthal-Zellerfeld , Germany"}]}],"member":"374","published-online":{"date-parts":[[2025,7,9]]},"reference":[{"key":"2025110513331642813_j_cmam-2024-0089_ref_001","doi-asserted-by":"crossref","unstructured":"B.  Andreianov, F.  Boyer and F.  Hubert,\nDiscrete duality finite volume schemes for Leray\u2013Lions-type elliptic problems on general 2D meshes,\nNumer. Methods Partial Differential Equations 23 (2007), no. 1, 145\u2013195.","DOI":"10.1002\/num.20170"},{"key":"2025110513331642813_j_cmam-2024-0089_ref_002","doi-asserted-by":"crossref","unstructured":"R.  Anton, D.  Cohen and L.  Quer-Sardanyons,\nA fully discrete approximation of the one-dimensional stochastic heat equation,\nIMA J. Numer. Anal. 40 (2020), no. 1, 247\u2013284.","DOI":"10.1093\/imanum\/dry060"},{"key":"2025110513331642813_j_cmam-2024-0089_ref_003","doi-asserted-by":"crossref","unstructured":"C.  Bauzet, E.  Bonetti, G.  Bonfanti, F.  Lebon and G.  Vallet,\nA global existence and uniqueness result for a stochastic Allen\u2013Cahn equation with constraint,\nMath. Methods Appl. Sci. 40 (2017), no. 14, 5241\u20135261.","DOI":"10.1002\/mma.4383"},{"key":"2025110513331642813_j_cmam-2024-0089_ref_004","doi-asserted-by":"crossref","unstructured":"C.  Bauzet, V.  Castel and J.  Charrier,\nExistence and uniqueness result for an hyperbolic scalar conservation law with a stochastic force using a finite volume approximation,\nJ. Hyperbolic Differ. Equ. 17 (2020), no. 2, 213\u2013294.","DOI":"10.1142\/S0219891620500071"},{"key":"2025110513331642813_j_cmam-2024-0089_ref_005","doi-asserted-by":"crossref","unstructured":"C.  Bauzet, J.  Charrier and T.  Gallou\u00ebt,\nConvergence of flux-splitting finite volume schemes for hyperbolic scalar conservation laws with a multiplicative stochastic perturbation,\nMath. Comp. 85 (2016), no. 302, 2777\u20132813.","DOI":"10.1090\/mcom\/3084"},{"key":"2025110513331642813_j_cmam-2024-0089_ref_006","doi-asserted-by":"crossref","unstructured":"C.  Bauzet, J.  Charrier and T.  Gallou\u00ebt,\nConvergence of monotone finite volume schemes for hyperbolic scalar conservation laws with multiplicative noise,\nStoch. Partial Differ. Equ. Anal. Comput. 4 (2016), no. 1, 150\u2013223.","DOI":"10.1007\/s40072-015-0052-z"},{"key":"2025110513331642813_j_cmam-2024-0089_ref_007","doi-asserted-by":"crossref","unstructured":"C.  Bauzet and F.  Nabet,\nConvergence of a finite-volume scheme for a heat equation with a multiplicative stochastic force,\nFinite Volumes for Complex Applications IX\u2014Methods, Theoretical Aspects, Examples\u2014FVCA,\nSpringer Proc. Math. Stat. 323,\nSpringer, Cham (2020), 275\u2013283.","DOI":"10.1007\/978-3-030-43651-3_24"},{"key":"2025110513331642813_j_cmam-2024-0089_ref_008","doi-asserted-by":"crossref","unstructured":"C.  Bauzet, F.  Nabet, K.  Schmitz and A.  Zimmermann,\nConvergence of a finite-volume scheme for a heat equation with a multiplicative Lipschitz noise,\nESAIM Math. Model. Numer. Anal. 57 (2023), no. 2, 745\u2013783.","DOI":"10.1051\/m2an\/2022087"},{"key":"2025110513331642813_j_cmam-2024-0089_ref_009","doi-asserted-by":"crossref","unstructured":"C.  Bauzet, F.  Nabet, K.  Schmitz and A.  Zimmermann,\nFinite volume approximations for non-linear parabolic problems with stochastic forcing,\nFinite Volumes for Complex Applications X. Vol. 1. Elliptic and Parabolic Problems,\nSpringer Proc. Math. Stat. 432,\nSpringer, Cham (2023), 157\u2013166.","DOI":"10.1007\/978-3-031-40864-9_10"},{"key":"2025110513331642813_j_cmam-2024-0089_ref_010","unstructured":"C.  Bauzet, K.  Schmitz and A.  Zimmermann,\nOn a finite-volume approximation of a diffusion-convection equation with a multiplicative stochastic force,\npreprint (2023), https:\/\/arxiv.org\/abs\/2304.02259."},{"key":"2025110513331642813_j_cmam-2024-0089_ref_011","doi-asserted-by":"crossref","unstructured":"M.  Bessemoulin-Chatard, C.  Chainais-Hillairet and F.  Filbet,\nOn discrete functional inequalities for some finite volume schemes,\nIMA J. Numer. Anal. 35 (2015), no. 3, 1125\u20131149.","DOI":"10.1093\/imanum\/dru032"},{"key":"2025110513331642813_j_cmam-2024-0089_ref_012","doi-asserted-by":"crossref","unstructured":"D.  Breit, M.  Hofmanov\u00e1 and S.  Loisel,\nSpace-time approximation of stochastic p-Laplace-type systems,\nSIAM J. Numer. Anal. 59 (2021), no. 4, 2218\u20132236.","DOI":"10.1137\/20M1334310"},{"key":"2025110513331642813_j_cmam-2024-0089_ref_013","doi-asserted-by":"crossref","unstructured":"D.  Breit and A.  Prohl,\nError analysis for 2D stochastic Navier\u2013Stokes equations in bounded domains with Dirichlet data,\nFound. Comput. Math. 24 (2024), no. 5, 1643\u20131672.","DOI":"10.1007\/s10208-023-09621-y"},{"key":"2025110513331642813_j_cmam-2024-0089_ref_014","doi-asserted-by":"crossref","unstructured":"H.  Brezis,\nFunctional Analysis, Sobolev Spaces and Partial Differential Equations,\nUniversitext,\nSpringer, New York, 2011.","DOI":"10.1007\/978-0-387-70914-7"},{"key":"2025110513331642813_j_cmam-2024-0089_ref_015","doi-asserted-by":"crossref","unstructured":"Y.  Coudi\u00e8re, J.-P.  Vila and P.  Villedieu,\nConvergence rate of a finite volume scheme for a two-dimensional convection-diffusion problem,\nM2AN Math. Model. Numer. Anal. 33 (1999), no. 3, 493\u2013516.","DOI":"10.1051\/m2an:1999149"},{"key":"2025110513331642813_j_cmam-2024-0089_ref_016","doi-asserted-by":"crossref","unstructured":"G.  Da Prato and J.  Zabczyk,\nStochastic Equations in Infinite Dimensions,\nEncyclopedia Math. Appl. 152,\nCambridge University, Cambridge, 2014.","DOI":"10.1017\/CBO9781107295513"},{"key":"2025110513331642813_j_cmam-2024-0089_ref_017","doi-asserted-by":"crossref","unstructured":"A.  Debussche and J.  Printems,\nWeak order for the discretization of the stochastic heat equation,\nMath. Comp. 78 (2009), no. 266, 845\u2013863.","DOI":"10.1090\/S0025-5718-08-02184-4"},{"key":"2025110513331642813_j_cmam-2024-0089_ref_018","doi-asserted-by":"crossref","unstructured":"L.  Diening, M.  Hofmanov\u00e1 and J.  Wichmann,\nAn averaged space-time discretization of the stochastic p-Laplace system,\nNumer. Math. 153 (2023), no. 2\u20133, 557\u2013609.","DOI":"10.1007\/s00211-022-01343-7"},{"key":"2025110513331642813_j_cmam-2024-0089_ref_019","doi-asserted-by":"crossref","unstructured":"J.  Droniou and R.  Eymard,\nA mixed finite volume scheme for anisotropic diffusion problems on any grid,\nNumer. Math. 105 (2006), no. 1, 35\u201371.","DOI":"10.1007\/s00211-006-0034-1"},{"key":"2025110513331642813_j_cmam-2024-0089_ref_020","doi-asserted-by":"crossref","unstructured":"R.  Eymard, T.  Gallou\u00ebt and R.  Herbin,\nFinite volume methods,\nSolution of Equations in \n                  \n                     \n                        \n                           \n                              \u211d\n                              n\n                           \n                        \n                        \n                        \\mathbb{R}^{n}\n                     \n                  \n               ,\nHandb. Numer. Anal. 7,\nNorth-Holland, Amsterdam (2000), 713\u20131020.","DOI":"10.1016\/S1570-8659(00)07005-8"},{"key":"2025110513331642813_j_cmam-2024-0089_ref_021","doi-asserted-by":"crossref","unstructured":"R.  Eymard, T.  Gallou\u00ebt and R.  Herbin,\nA cell-centered finite-volume approximation for anisotropic diffusion operators on unstructured meshes in any space dimension,\nIMA J. Numer. Anal. 26 (2006), no. 2, 326\u2013353.","DOI":"10.1093\/imanum\/dri036"},{"key":"2025110513331642813_j_cmam-2024-0089_ref_022","doi-asserted-by":"crossref","unstructured":"X.  Feng and H.  Qiu,\nAnalysis of fully discrete mixed finite element methods for time-dependent stochastic Stokes equations with multiplicative noise,\nJ. Sci. Comput. 88 (2021), no. 2, Paper No. 31.","DOI":"10.1007\/s10915-021-01546-4"},{"key":"2025110513331642813_j_cmam-2024-0089_ref_023","unstructured":"P.  Grisvard,\nElliptic Problems in Nonsmooth Domains,\nMonogr. Stud. Math. 24,\nPitman, Boston, 1985."},{"key":"2025110513331642813_j_cmam-2024-0089_ref_024","doi-asserted-by":"crossref","unstructured":"I.  Gy\u00f6ngy and A.  Millet,\nOn discretization schemes for stochastic evolution equations,\nPotential Anal. 23 (2005), no. 2, 99\u2013134.","DOI":"10.1007\/s11118-004-5393-6"},{"key":"2025110513331642813_j_cmam-2024-0089_ref_025","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":"2025110513331642813_j_cmam-2024-0089_ref_026","doi-asserted-by":"crossref","unstructured":"U.  Koley and G.  Vallet,\nOn the rate of convergence of a numerical scheme for fractional conservation laws with noise,\nIMA J. Numer. Anal. 44 (2024), no. 3, 1372\u20131405.","DOI":"10.1093\/imanum\/drad015"},{"key":"2025110513331642813_j_cmam-2024-0089_ref_027","doi-asserted-by":"crossref","unstructured":"N. V.  Krylov and B. L.  Rozovski\u012d,\nStochastic evolution equations,\nJ. Soviet Math. 16 (1981), no. 4, 1233\u20131277.","DOI":"10.1007\/BF01084893"},{"key":"2025110513331642813_j_cmam-2024-0089_ref_028","doi-asserted-by":"crossref","unstructured":"W.  Liu and M.  R\u00f6ckner,\nStochastic Partial Differential Equations: An Introduction,\nUniversitext,\nSpringer, Cham, 2015.","DOI":"10.1007\/978-3-319-22354-4"},{"key":"2025110513331642813_j_cmam-2024-0089_ref_029","doi-asserted-by":"crossref","unstructured":"A. K.  Majee and A.  Prohl,\nOptimal strong rates of convergence for a space-time discretization of the stochastic Allen\u2013Cahn equation with multiplicative noise,\nComput. Methods Appl. Math. 18 (2018), no. 2, 297\u2013311.","DOI":"10.1515\/cmam-2017-0023"},{"key":"2025110513331642813_j_cmam-2024-0089_ref_030","doi-asserted-by":"crossref","unstructured":"F.  Nabet,\nAn error estimate for a finite-volume scheme for the Cahn\u2013Hilliard equation with dynamic boundary conditions,\nNumer. Math. 149 (2021), no. 1, 185\u2013226.","DOI":"10.1007\/s00211-021-01230-7"},{"key":"2025110513331642813_j_cmam-2024-0089_ref_031","doi-asserted-by":"crossref","unstructured":"M.  Ondrej\u00e1t, A.  Prohl and N. J.  Walkington,\nNumerical approximation of nonlinear SPDE\u2019s,\nStoch. Partial Differ. Equ. Anal. Comput. 11 (2023), no. 4, 1553\u20131634.","DOI":"10.1007\/s40072-022-00271-9"},{"key":"2025110513331642813_j_cmam-2024-0089_ref_032","unstructured":"\u00c9.  Pardoux,\n\u00c9quations aux d\u00e9riv\u00e9es partielles stochastiques non lin\u00e9aires monotones,\nPhd thesis, University Paris Sud, 1975."},{"key":"2025110513331642813_j_cmam-2024-0089_ref_033","doi-asserted-by":"crossref","unstructured":"N.  Sapountzoglou, P.  Wittbold and A.  Zimmermann,\nOn a doubly nonlinear PDE with stochastic perturbation,\nStoch. Partial Differ. Equ. Anal. Comput. 7 (2019), no. 2, 297\u2013330.","DOI":"10.1007\/s40072-018-0128-7"},{"key":"2025110513331642813_j_cmam-2024-0089_ref_034","doi-asserted-by":"crossref","unstructured":"S.  Sugiyama,\nStability problems on difference and functional-differential equations,\nProc. Japan Acad. 45 (1969), 526\u2013529.","DOI":"10.3792\/pja\/1195520661"}],"container-title":["Computational Methods in Applied Mathematics"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.degruyterbrill.com\/document\/doi\/10.1515\/cmam-2024-0089\/xml","content-type":"application\/xml","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.degruyterbrill.com\/document\/doi\/10.1515\/cmam-2024-0089\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,11,5]],"date-time":"2025-11-05T13:35:52Z","timestamp":1762349752000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.degruyterbrill.com\/document\/doi\/10.1515\/cmam-2024-0089\/html"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,7,9]]},"references-count":34,"journal-issue":{"issue":"4","published-online":{"date-parts":[[2025,6,25]]},"published-print":{"date-parts":[[2025,10,1]]}},"alternative-id":["10.1515\/cmam-2024-0089"],"URL":"https:\/\/doi.org\/10.1515\/cmam-2024-0089","relation":{},"ISSN":["1609-4840","1609-9389"],"issn-type":[{"type":"print","value":"1609-4840"},{"type":"electronic","value":"1609-9389"}],"subject":[],"published":{"date-parts":[[2025,7,9]]}}}