{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,5,13]],"date-time":"2026-05-13T22:14:52Z","timestamp":1778710492077,"version":"3.51.4"},"reference-count":49,"publisher":"Springer Science and Business Media LLC","issue":"2","license":[{"start":{"date-parts":[[2026,2,19]],"date-time":"2026-02-19T00:00:00Z","timestamp":1771459200000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by-nc-nd\/4.0"},{"start":{"date-parts":[[2026,2,19]],"date-time":"2026-02-19T00:00:00Z","timestamp":1771459200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by-nc-nd\/4.0"}],"funder":[{"name":"NTNU Norwegian University of Science and Technology"}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Adv Comput Math"],"published-print":{"date-parts":[[2026,4]]},"DOI":"10.1007\/s10444-026-10291-x","type":"journal-article","created":{"date-parts":[[2026,2,19]],"date-time":"2026-02-19T09:09:21Z","timestamp":1771492161000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Spectral approximation of a class of stochastic time-fractional evolution equations"],"prefix":"10.1007","volume":"52","author":[{"ORCID":"https:\/\/orcid.org\/0009-0004-6464-8006","authenticated-orcid":false,"given":"Simen Knutsen","family":"Furset","sequence":"first","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2026,2,19]]},"reference":[{"key":"10291_CR1","doi-asserted-by":"publisher","first-page":"1335","DOI":"10.1090\/mcom\/3016","volume":"85","author":"A Andersson","year":"2015","unstructured":"Andersson, A., Larsson, S.: Weak convergence for a spatial approximation of the nonlinear stochastic heat equation. Math. Comput. 85, 1335\u20131358 (2015)","journal-title":"Math. Comput."},{"key":"10291_CR2","doi-asserted-by":"publisher","first-page":"3","DOI":"10.1007\/s00477-007-0157-7","volume":"22","author":"JM Angulo","year":"2007","unstructured":"Angulo, J.M., Kelbert, M., Leonenko, N., Ruiz-Medina, M.: Spatiotemporal random fields associated with stochastic fractional Helmholtz and heat equations. Stoch. Env. Res. Risk Assess. 22, 3\u201313 (2007)","journal-title":"Stoch. Env. Res. Risk Assess."},{"key":"10291_CR3","doi-asserted-by":"publisher","first-page":"497","DOI":"10.1007\/s11118-021-09976-3","volume":"59","author":"L Ba\u0148as","year":"2022","unstructured":"Ba\u0148as, L., Yang, H., Zhu, R.: Sharp interface limit of stochastic Cahn-Hilliard equation with singular noise. Potential Analysis 59, 497\u2013518 (2022)","journal-title":"Potential Analysis"},{"key":"10291_CR4","doi-asserted-by":"crossref","unstructured":"Bakka, H., Rue, H., Fuglstad, G., Riebler, A., Bolin, D., Illian, J., Krainski, E., Simpson, D., Lindgren, F.: Spatial modeling with R-INLA: a review, WIREs Computational Statistics, 10 (2018)","DOI":"10.1002\/wics.1443"},{"key":"10291_CR5","first-page":"64","volume":"73","author":"P Beesack","year":"1966","unstructured":"Beesack, P.: A general form of the remainder in Taylor\u2019s theorem. Am. Math. Mon. 73, 64 (1966)","journal-title":"Am. Math. Mon."},{"key":"10291_CR6","doi-asserted-by":"publisher","first-page":"2319","DOI":"10.1137\/20M1339520","volume":"53","author":"A Biswas","year":"2021","unstructured":"Biswas, A., De Le\u00f3n-Contreras, M., Stinga, P.: Harnack inequalities and H\u00f6lder estimates for master equations. SIAM J. Math. Anal. 53, 2319\u20132348 (2021)","journal-title":"SIAM J. Math. Anal."},{"key":"10291_CR7","doi-asserted-by":"publisher","first-page":"503","DOI":"10.1007\/s00028-020-00590-1","volume":"21","author":"A Biswas","year":"2020","unstructured":"Biswas, A., Stinga, P.: Regularity estimates for nonlocal space-time master equations in bounded domains. J. Evol. Equ. 21, 503\u2013565 (2020)","journal-title":"J. Evol. Equ."},{"key":"10291_CR8","doi-asserted-by":"publisher","first-page":"274","DOI":"10.1080\/10618600.2019.1665537","volume":"29","author":"D Bolin","year":"2019","unstructured":"Bolin, D., Kirchner, K.: The rational SPDE approach for gaussian random fields with general smoothness. J. Comput. Graph. Stat. 29, 274\u2013285 (2019)","journal-title":"J. Comput. Graph. Stat."},{"key":"10291_CR9","doi-asserted-by":"publisher","first-page":"1051","DOI":"10.1093\/imanum\/dry091","volume":"40","author":"D Bolin","year":"2018","unstructured":"Bolin, D., Kirchner, K., Kov\u00e1cs, M.: Numerical solution of fractional elliptic stochastic PDEs with spatial white noise. IMA J. Numer. Anal. 40, 1051\u20131073 (2018)","journal-title":"IMA J. Numer. Anal."},{"key":"10291_CR10","doi-asserted-by":"publisher","first-page":"881","DOI":"10.1007\/s10543-018-0719-8","volume":"58","author":"D Bolin","year":"2018","unstructured":"Bolin, D., Kirchner, K., Kov\u00e1cs, M.: Weak convergence of Galerkin approximations for fractional elliptic stochastic PDEs with spatial white noise. BIT Numer. Math. 58, 881\u2013906 (2018)","journal-title":"BIT Numer. Math."},{"key":"10291_CR11","doi-asserted-by":"crossref","unstructured":"Bonaccorsi, S.: Fractional stochastic evolution equations with L\u00e9vy noise. Diff. Integral Equ., 22 (2009)","DOI":"10.57262\/die\/1356019409"},{"key":"10291_CR12","doi-asserted-by":"publisher","first-page":"57","DOI":"10.1515\/jnma-2017-0116","volume":"27","author":"A Bonito","year":"2019","unstructured":"Bonito, A., Lei, W., Pasciak, J.: On sinc quadrature approximations of fractional powers of regularly accretive operators. J. Numer. Math. 27, 57\u201368 (2019)","journal-title":"J. Numer. Math."},{"key":"10291_CR13","doi-asserted-by":"publisher","DOI":"10.1007\/s40072-024-00326-z","volume-title":"Weak error analysis for the stochastic Allen\u2013Cahn equation","author":"D Breit","year":"2024","unstructured":"Breit, D., Prohl, A.: Weak error analysis for the stochastic Allen\u2013Cahn equation. Analysis and Computations, Stochastics and Partial Differential Equations (2024)"},{"key":"10291_CR14","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1007\/s11118-013-9338-9","volume":"40","author":"C Br\u00e9hier","year":"2013","unstructured":"Br\u00e9hier, C.: Approximation of the invariant measure with an Euler scheme for stochastic PDEs driven by space-time white noise. Potential Analysis 40, 1\u201340 (2013)","journal-title":"Potential Analysis"},{"key":"10291_CR15","unstructured":"Br\u00e9hier, C., Cui, J., Wang, X.: Weak error estimates of fully-discrete schemes for the stochastic Cahn-Hilliard equation, (2022)"},{"key":"10291_CR16","doi-asserted-by":"crossref","unstructured":"Cohen, D., Lang, A.: Numerical approximation and simulation of the stochastic wave equation on the sphere, Calcolo, 59 (2022)","DOI":"10.1007\/s10092-022-00472-7"},{"key":"10291_CR17","volume-title":"Statistics for Spatio-Temporal Data","author":"N Cressie","year":"2011","unstructured":"Cressie, N., Wikle, C.: Statistics for Spatio-Temporal Data. Wiley-Blackwell, Chichester, England (2011)"},{"key":"10291_CR18","doi-asserted-by":"crossref","unstructured":"Da Prato, G., Zabczyk, J.: Stochastic equations in infinite dimensions, Cambridge University Press, Apr. (2014)","DOI":"10.1017\/CBO9781107295513"},{"key":"10291_CR19","doi-asserted-by":"crossref","unstructured":"Davies, E.: Spectral theory and differential operators, Cambridge University Press, Jan. (1995)","DOI":"10.1017\/CBO9780511623721"},{"key":"10291_CR20","doi-asserted-by":"publisher","first-page":"200","DOI":"10.1016\/S0167-2789(99)00072-X","volume":"134","author":"A Debussche","year":"1999","unstructured":"Debussche, A., Printems, J.: Numerical simulation of the stochastic Korteweg\u2013de Vries equation. Physica D 134, 200\u2013226 (1999)","journal-title":"Physica D"},{"key":"10291_CR21","doi-asserted-by":"publisher","first-page":"2305","DOI":"10.3934\/cpaa.2012.11.2305","volume":"11","author":"A Debussche","year":"2011","unstructured":"Debussche, A., Vovelle, J.: Diffusion limit for a stochastic kinetic problem. Communications on Pure and Applied Analysis 11, 2305\u20132326 (2011)","journal-title":"Communications on Pure and Applied Analysis"},{"key":"10291_CR22","doi-asserted-by":"crossref","unstructured":"Desch, W., Londen, S.: Semilinear stochastic integral equations in $$L^p$$, Springer Basel, pp. 131\u2013166 (2011)","DOI":"10.1007\/978-3-0348-0075-4_8"},{"key":"10291_CR23","doi-asserted-by":"publisher","first-page":"1044","DOI":"10.1007\/s40072-022-00250-0","volume":"11","author":"K Fahim","year":"2022","unstructured":"Fahim, K., Hausenblas, E., Kov\u00e1cs, M.: Some approximation results for mild solutions of stochastic fractional order evolution equations driven by Gaussian noise. Stochastics and Partial Differential Equations: Analysis and Computations 11, 1044\u20131088 (2022)","journal-title":"Stochastics and Partial Differential Equations: Analysis and Computations"},{"key":"10291_CR24","doi-asserted-by":"crossref","unstructured":"Fuglstad, D., Lindgren, F., Simpson, D., Rue, H.: Exploring a new class of non-stationary spatial Gaussian random fields with varying local anisotropy. Statistica Sinica, (2014)","DOI":"10.5705\/ss.2013.106w"},{"key":"10291_CR25","doi-asserted-by":"publisher","first-page":"A825","DOI":"10.1137\/21M1400717","volume":"44","author":"E Jansson","year":"2022","unstructured":"Jansson, E., Kov\u00e1cs, M., Lang, A.: Surface finite element approximation of spherical Whittle-Mat\u00e9rn Gaussian random fields. SIAM J. Sci. Comput. 44, A825\u2013A842 (2022)","journal-title":"SIAM J. Sci. Comput."},{"key":"10291_CR26","doi-asserted-by":"crossref","unstructured":"Jentzen, A., Kloeden, P.: Taylor approximations for stochastic partial differential equations. Soc. Indust. Appl. Math., Jan. (2011)","DOI":"10.1137\/1.9781611972016"},{"key":"10291_CR27","doi-asserted-by":"publisher","first-page":"108","DOI":"10.1239\/aap\/1113402402","volume":"37","author":"M Kelbert","year":"2005","unstructured":"Kelbert, M., Leonenko, N., Ruiz-Medina, M.: Fractional random fields associated with stochastic fractional heat equations. Adv. Appl. Probab. 37, 108\u2013133 (2005)","journal-title":"Adv. Appl. Probab."},{"key":"10291_CR28","doi-asserted-by":"publisher","first-page":"5896","DOI":"10.1016\/j.jde.2017.02.021","volume":"262","author":"K Kirchner","year":"2017","unstructured":"Kirchner, K., Lang, A., Larsson, S.: Covariance structure of parabolic stochastic partial differential equations with multiplicative L\u00e9vy noise. J. Differential Equations 262, 5896\u20135927 (2017)","journal-title":"J. Differential Equations"},{"key":"10291_CR29","doi-asserted-by":"publisher","first-page":"1805","DOI":"10.1007\/s40072-023-00316-7","volume":"12","author":"K Kirchner","year":"2023","unstructured":"Kirchner, K., Willems, J.: Regularity theory for a new class of fractional parabolic stochastic evolution equations. Stochastics and Partial Differential Equations: Analysis and Computations 12, 1805\u20131854 (2023)","journal-title":"Stochastics and Partial Differential Equations: Analysis and Computations"},{"key":"10291_CR30","doi-asserted-by":"crossref","unstructured":"Kirchner, K., Willems, J.: Multiple and weak Markov properties in Hilbert spaces with applications to fractional stochastic evolution equations. Stochastic Processes and their Applications 186,(2025)","DOI":"10.1016\/j.spa.2025.104639"},{"key":"10291_CR31","doi-asserted-by":"publisher","first-page":"1324","DOI":"10.1093\/imanum\/drac020","volume":"43","author":"M Kov\u00e1cs","year":"2022","unstructured":"Kov\u00e1cs, M., Lang, A., Petersson, A.: Approximation of spde covariance operators by finite elements: a semigroup approach. IMA J. Numer. Anal. 43, 1324\u20131357 (2022)","journal-title":"IMA J. Numer. Anal."},{"key":"10291_CR32","unstructured":"Kov\u00e1cs, M., Larsson, S.: Introduction to stochastic partial differential, (2010)"},{"key":"10291_CR33","doi-asserted-by":"publisher","first-page":"309","DOI":"10.1007\/s11075-009-9281-4","volume":"53","author":"M Kov\u00e1cs","year":"2009","unstructured":"Kov\u00e1cs, M., Larsson, S., Lindgren, F.: Strong convergence of the finite element method with truncated noise for semilinear parabolic stochastic equations with additive noise. Numerical Algorithms 53, 309\u2013320 (2009)","journal-title":"Numerical Algorithms"},{"key":"10291_CR34","doi-asserted-by":"publisher","first-page":"85","DOI":"10.1007\/s10543-011-0344-2","volume":"52","author":"M Kov\u00e1cs","year":"2011","unstructured":"Kov\u00e1cs, M., Larsson, S., Lindgren, F.: Weak convergence of finite element approximations of linear stochastic evolution equations with additive noise. BIT Numer. Math. 52, 85\u2013108 (2011)","journal-title":"BIT Numer. Math."},{"key":"10291_CR35","doi-asserted-by":"publisher","first-page":"66","DOI":"10.1137\/18M1177895","volume":"58","author":"M Kov\u00e1cs","year":"2020","unstructured":"Kov\u00e1cs, M., Larsson, S., Saedpanah, F.: Mittag-leffler euler integrator for a stochastic fractional order equation with additive noise. SIAM J. Numer. Anal. 58, 66\u201385 (2020)","journal-title":"SIAM J. Numer. Anal."},{"key":"10291_CR36","doi-asserted-by":"publisher","DOI":"10.4171\/151","volume-title":"Spectral Theory in Riemannian Geometry","author":"O Labl\u00e9e","year":"2015","unstructured":"Labl\u00e9e, O.: Spectral Theory in Riemannian Geometry. EMS Press, Zuerich, Switzerland (2015)"},{"key":"10291_CR37","first-page":"23","volume":"11","author":"A Lang","year":"2024","unstructured":"Lang, A., Motschan-Armen, I.: Euler\u2013maruyama approximations of the stochastic heat equation on the sphere, Journal of Computational. Dynamics 11, 23\u201342 (2024)","journal-title":"Dynamics"},{"key":"10291_CR38","doi-asserted-by":"crossref","unstructured":"Lang, A., Schwab, C.: Isotropic gaussian random fields on the sphere: Regularity, fast simulation and stochastic partial differential equations. The Annals of Applied Probability, 25 (2015)","DOI":"10.1214\/14-AAP1067"},{"key":"10291_CR39","first-page":"3","volume":"48","author":"F Lindgren","year":"2024","unstructured":"Lindgren, F., Bakka, H., Bolin, D., Krainski, E., Rue, H.: A diffusion-based spatio-temporal extension of gaussian mat\u00e9rn fields. SORT 48, 3\u201366 (2024)","journal-title":"SORT"},{"key":"10291_CR40","doi-asserted-by":"publisher","DOI":"10.1016\/j.spasta.2022.100599","volume":"50","author":"F Lindgren","year":"2022","unstructured":"Lindgren, F., Bolin, D., Rue, H.: The spde approach for gaussian and non-gaussian fields: 10 years and still running. Spatial Statistics 50, 100599 (2022)","journal-title":"Spatial Statistics"},{"key":"10291_CR41","first-page":"423","volume":"73","author":"F Lindgren","year":"2011","unstructured":"Lindgren, F., Rue, H., Lindstr\u00f6m, J.: An explicit link between gaussian fields and gaussian markov random fields: The stochastic partial differential equation approach, Journal of the Royal Statistical Society Series B. Statistical Methodology 73, 423\u2013498 (2011)","journal-title":"Statistical Methodology"},{"key":"10291_CR42","doi-asserted-by":"crossref","unstructured":"Litsg\u00e5rd, M., Nystr\u00f6m, K.: On local regularity estimates for fractional powers of parabolic operators with time-dependent measurable coefficients. J. Evol. Equ., 23 (2022)","DOI":"10.1007\/s00028-022-00844-0"},{"key":"10291_CR43","doi-asserted-by":"crossref","unstructured":"Lord, G., Powell, C., Shardlow, T.: An Introduction to Computational Stochastic PDEs, Cambridge University Press, June (2014)","DOI":"10.1017\/CBO9781139017329"},{"key":"10291_CR44","first-page":"29","volume":"140","author":"K Nystr\u00f6m","year":"2016","unstructured":"Nystr\u00f6m, K., Sande, O.: Extension properties and boundary estimates for a fractional heat operator, Nonlinear. Analysis 140, 29\u201337 (2016)","journal-title":"Analysis"},{"key":"10291_CR45","doi-asserted-by":"publisher","first-page":"3386","DOI":"10.1093\/imanum\/drab069","volume":"42","author":"A Prohl","year":"2021","unstructured":"Prohl, A., Wang, Y.: Strong error estimates for a space-time discretization of the linear-quadratic control problem with the stochastic heat equation with linear noise. IMA J. Numer. Anal. 42, 3386\u20133429 (2021)","journal-title":"IMA J. Numer. Anal."},{"key":"10291_CR46","doi-asserted-by":"publisher","first-page":"3893","DOI":"10.1137\/16M1104317","volume":"49","author":"P Stinga","year":"2017","unstructured":"Stinga, P., Torrea, J.: Regularity theory and extension problem for fractional nonlocal parabolic equations and the master equation. SIAM J. Math. Anal. 49, 3893\u20133924 (2017)","journal-title":"SIAM J. Math. Anal."},{"key":"10291_CR47","volume-title":"An analysis of the finite element method","author":"G Strang","year":"1969","unstructured":"Strang, G., Fix, G.: An analysis of the finite element method. Prentice-Hall, Englewood Cliffs (1969)"},{"key":"10291_CR48","doi-asserted-by":"crossref","unstructured":"Willems, J.: Dirichlet problems associated to abstract nonlocal space\u2013time differential operators. J. Evol. Equ., 25 (2025)","DOI":"10.1007\/s00028-024-01026-w"},{"key":"10291_CR49","doi-asserted-by":"publisher","first-page":"1363","DOI":"10.1137\/040605278","volume":"43","author":"Y Yan","year":"2005","unstructured":"Yan, Y.: Galerkin finite element methods for stochastic parabolic partial differential equations. SIAM J. Numer. Anal. 43, 1363\u20131384 (2005)","journal-title":"SIAM J. Numer. Anal."}],"container-title":["Advances in Computational Mathematics"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s10444-026-10291-x.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s10444-026-10291-x","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s10444-026-10291-x.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,5,13]],"date-time":"2026-05-13T22:02:56Z","timestamp":1778709776000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s10444-026-10291-x"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2026,2,19]]},"references-count":49,"journal-issue":{"issue":"2","published-print":{"date-parts":[[2026,4]]}},"alternative-id":["10291"],"URL":"https:\/\/doi.org\/10.1007\/s10444-026-10291-x","relation":{},"ISSN":["1019-7168","1572-9044"],"issn-type":[{"value":"1019-7168","type":"print"},{"value":"1572-9044","type":"electronic"}],"subject":[],"published":{"date-parts":[[2026,2,19]]},"assertion":[{"value":"7 July 2025","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"3 February 2026","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"19 February 2026","order":3,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}},{"order":1,"name":"Ethics","group":{"name":"EthicsHeading","label":"Declarations"}},{"value":"The author declares no competing interests.","order":2,"name":"Ethics","group":{"name":"EthicsHeading","label":"Conflict of interest"}}],"article-number":"16"}}