--- DOAJ_listed: '1' OA_place: publisher OA_type: gold PlanS_conform: '1' _id: '18706' abstract: - lang: eng text: "We prove discrete-to-continuum convergence for dynamical optimal transport on Zd\r\n -periodic graphs with cost functional having linear growth at infinity. This result provides an answer to a problem left open by Gladbach, Kopfer, Maas, and Portinale (Calc Var Partial Differential Equations 62(5), 2023), where the convergence behaviour of discrete boundary-value dynamical transport problems is proved under the stronger assumption of superlinear growth. Our result extends the known literature to some important classes of examples, such as scaling limits of 1 -Wasserstein transport problems. Similarly to what happens in the quadratic case, the geometry of the graph plays a crucial role in the structure of the limit cost function, as we discuss in the final part of this work, which includes some visual representations." acknowledgement: L.P. gratefully acknowledges fundings from the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under Germany’s Excellence Strategy – GZ 2047/1, Projekt-ID 390685813. F.Q. gratefully acknowledges support from the Austrian Science Fund (FWF) project 10.55776/F65. article_processing_charge: Yes article_type: original author: - first_name: Lorenzo full_name: Portinale, Lorenzo id: 30AD2CBC-F248-11E8-B48F-1D18A9856A87 last_name: Portinale - first_name: Filippo full_name: Quattrocchi, Filippo id: 3ebd6ba8-edfb-11eb-afb5-91a9745ba308 last_name: Quattrocchi orcid: 0009-0000-9773-1931 citation: ama: Portinale L, Quattrocchi F. Discrete-to-continuum limits of optimal transport with linear growth on periodic graphs. European Journal of Applied Mathematics. 2026;37(3):614-642. doi:10.1017/s0956792524000810 apa: Portinale, L., & Quattrocchi, F. (2026). Discrete-to-continuum limits of optimal transport with linear growth on periodic graphs. European Journal of Applied Mathematics. Cambridge University Press. https://doi.org/10.1017/s0956792524000810 chicago: Portinale, Lorenzo, and Filippo Quattrocchi. “Discrete-to-Continuum Limits of Optimal Transport with Linear Growth on Periodic Graphs.” European Journal of Applied Mathematics. Cambridge University Press, 2026. https://doi.org/10.1017/s0956792524000810. ieee: L. Portinale and F. Quattrocchi, “Discrete-to-continuum limits of optimal transport with linear growth on periodic graphs,” European Journal of Applied Mathematics, vol. 37, no. 3. Cambridge University Press, pp. 614–642, 2026. ista: Portinale L, Quattrocchi F. 2026. Discrete-to-continuum limits of optimal transport with linear growth on periodic graphs. European Journal of Applied Mathematics. 37(3), 614–642. mla: Portinale, Lorenzo, and Filippo Quattrocchi. “Discrete-to-Continuum Limits of Optimal Transport with Linear Growth on Periodic Graphs.” European Journal of Applied Mathematics, vol. 37, no. 3, Cambridge University Press, 2026, pp. 614–42, doi:10.1017/s0956792524000810. short: L. Portinale, F. Quattrocchi, European Journal of Applied Mathematics 37 (2026) 614–642. das_tickbox: '0' date_created: 2024-12-23T11:03:59Z date_published: 2026-06-01T00:00:00Z date_updated: 2026-10-27T23:31:03Z day: '01' ddc: - '500' department: - _id: GradSch - _id: JaMa doi: 10.1017/s0956792524000810 external_id: isi: - '001381435800001' file: - access_level: open_access checksum: d038f4d00cbfbde2672c17138eab21c9 content_type: application/pdf creator: dernst date_created: 2026-07-23T05:55:03Z date_updated: 2026-07-23T05:55:03Z file_id: '22386' file_name: 2026_EuropJourAppliedMath_Portinale.pdf file_size: 612317 relation: main_file success: 1 file_date_updated: 2026-07-23T05:55:03Z fulldoi: https://doi.org/10.1017/s0956792524000810 has_accepted_license: '1' intvolume: ' 37' isi: 1 issue: '3' keyword: - optimal transport - discrete-to-continuum - homogenisation - linear growth - gamma-convergence language: - iso: eng license: https://creativecommons.org/licenses/by/4.0/ month: '06' oa: 1 oa_version: Published Version page: 614-642 project: - _id: fc31cba2-9c52-11eb-aca3-ff467d239cd2 grant_number: F6504 name: Taming Complexity in Partial Differential Systems publication: European Journal of Applied Mathematics publication_identifier: eissn: - 1469-4425 issn: - 0956-7925 publication_status: published publisher: Cambridge University Press quality_controlled: '1' related_material: record: - id: '20563' relation: dissertation_contains status: public researchdata_availability: no scopus_import: '1' status: public supplementarymaterial: no title: Discrete-to-continuum limits of optimal transport with linear growth on periodic graphs tmp: image: /images/cc_by.png legal_code_url: https://creativecommons.org/licenses/by/4.0/legalcode name: Creative Commons Attribution 4.0 International Public License (CC-BY 4.0) short: CC BY (4.0) type: journal_article user_id: 317138e5-6ab7-11ef-aa6d-ffef3953e345 volume: 37 year: '2026' ... --- _id: '11739' abstract: - lang: eng text: We consider finite-volume approximations of Fokker--Planck equations on bounded convex domains in $\mathbb{R}^d$ and study the corresponding gradient flow structures. We reprove the convergence of the discrete to continuous Fokker--Planck equation via the method of evolutionary $\Gamma$-convergence, i.e., we pass to the limit at the level of the gradient flow structures, generalizing the one-dimensional result obtained by Disser and Liero. The proof is of variational nature and relies on a Mosco convergence result for functionals in the discrete-to-continuum limit that is of independent interest. Our results apply to arbitrary regular meshes, even though the associated discrete transport distances may fail to converge to the Wasserstein distance in this generality. acknowledgement: This work was supported by the European Research Council (ERC) under the European Union's Horizon 2020 Research and Innovation Programme grant 716117 and by the AustrianScience Fund (FWF) through grants F65 and W1245. article_processing_charge: No article_type: original arxiv: 1 author: - first_name: Dominik L full_name: Forkert, Dominik L id: 35C79D68-F248-11E8-B48F-1D18A9856A87 last_name: Forkert - first_name: Jan full_name: Maas, Jan id: 4C5696CE-F248-11E8-B48F-1D18A9856A87 last_name: Maas orcid: 0000-0002-0845-1338 - first_name: Lorenzo full_name: Portinale, Lorenzo id: 30AD2CBC-F248-11E8-B48F-1D18A9856A87 last_name: Portinale citation: ama: Forkert DL, Maas J, Portinale L. Evolutionary $\Gamma$-convergence of entropic gradient flow structures for Fokker-Planck equations in multiple dimensions. SIAM Journal on Mathematical Analysis. 2022;54(4):4297-4333. doi:10.1137/21M1410968 apa: Forkert, D. L., Maas, J., & Portinale, L. (2022). Evolutionary $\Gamma$-convergence of entropic gradient flow structures for Fokker-Planck equations in multiple dimensions. SIAM Journal on Mathematical Analysis. Society for Industrial and Applied Mathematics. https://doi.org/10.1137/21M1410968 chicago: Forkert, Dominik L, Jan Maas, and Lorenzo Portinale. “Evolutionary $\Gamma$-Convergence of Entropic Gradient Flow Structures for Fokker-Planck Equations in Multiple Dimensions.” SIAM Journal on Mathematical Analysis. Society for Industrial and Applied Mathematics, 2022. https://doi.org/10.1137/21M1410968. ieee: D. L. Forkert, J. Maas, and L. Portinale, “Evolutionary $\Gamma$-convergence of entropic gradient flow structures for Fokker-Planck equations in multiple dimensions,” SIAM Journal on Mathematical Analysis, vol. 54, no. 4. Society for Industrial and Applied Mathematics, pp. 4297–4333, 2022. ista: Forkert DL, Maas J, Portinale L. 2022. Evolutionary $\Gamma$-convergence of entropic gradient flow structures for Fokker-Planck equations in multiple dimensions. SIAM Journal on Mathematical Analysis. 54(4), 4297–4333. mla: Forkert, Dominik L., et al. “Evolutionary $\Gamma$-Convergence of Entropic Gradient Flow Structures for Fokker-Planck Equations in Multiple Dimensions.” SIAM Journal on Mathematical Analysis, vol. 54, no. 4, Society for Industrial and Applied Mathematics, 2022, pp. 4297–333, doi:10.1137/21M1410968. short: D.L. Forkert, J. Maas, L. Portinale, SIAM Journal on Mathematical Analysis 54 (2022) 4297–4333. corr_author: '1' date_created: 2022-08-07T22:01:59Z date_published: 2022-07-18T00:00:00Z date_updated: 2025-04-15T08:31:31Z day: '18' department: - _id: JaMa doi: 10.1137/21M1410968 ec_funded: 1 external_id: arxiv: - '2008.10962' isi: - '000889274600001' fulldoi: https://doi.org/10.1137/21M1410968 intvolume: ' 54' isi: 1 issue: '4' keyword: - Fokker--Planck equation - gradient flow - evolutionary $\Gamma$-convergence language: - iso: eng main_file_link: - open_access: '1' url: ' https://doi.org/10.48550/arXiv.2008.10962' month: '07' oa: 1 oa_version: Preprint page: 4297-4333 project: - _id: 256E75B8-B435-11E9-9278-68D0E5697425 call_identifier: H2020 grant_number: '716117' name: Optimal Transport and Stochastic Dynamics - _id: fc31cba2-9c52-11eb-aca3-ff467d239cd2 grant_number: F6504 name: Taming Complexity in Partial Differential Systems - _id: 260788DE-B435-11E9-9278-68D0E5697425 call_identifier: FWF grant_number: W1245 name: Dissipation and dispersion in nonlinear partial differential equations publication: SIAM Journal on Mathematical Analysis publication_identifier: eissn: - 1095-7154 issn: - 0036-1410 publication_status: published publisher: Society for Industrial and Applied Mathematics quality_controlled: '1' related_material: record: - id: '10022' relation: earlier_version status: public scopus_import: '1' status: public title: Evolutionary $\Gamma$-convergence of entropic gradient flow structures for Fokker-Planck equations in multiple dimensions type: journal_article user_id: 4359f0d1-fa6c-11eb-b949-802e58b17ae8 volume: 54 year: '2022' ... --- OA_type: closed access _id: '1315' abstract: - lang: eng text: We prove optimal second order convergence of a modified lowest-order Brezzi-Douglas-Marini (BDM1) mixed finite element scheme for advection-diffusion problems in divergence form. If advection is present, it is known that the total flux is approximated only with first-order accuracy by the classical BDM1 mixed method, which is suboptimal since the same order of convergence is obtained if the computationally less expensive Raviart-Thomas (RT0) element is used. The modification that was first proposed by Brunner et al. [Adv. Water Res., 35 (2012),pp. 163-171] is based on the hybrid problem formulation and consists in using the Lagrange multipliers for the discretization of the advective term instead of the cellwise constant approximation of the scalar unknown. article_processing_charge: No article_type: original author: - first_name: Fabian full_name: Brunner, Fabian last_name: Brunner - first_name: Julian L full_name: Fischer, Julian L id: 2C12A0B0-F248-11E8-B48F-1D18A9856A87 last_name: Fischer orcid: 0000-0002-0479-558X - first_name: Peter full_name: Knabner, Peter last_name: Knabner citation: ama: Brunner F, Fischer JL, Knabner P. Analysis of a modified second-order mixed hybrid BDM1 finite element method for transport problems in divergence form. Journal on Numerical Analysis. 2016;54(4):2359-2378. doi:10.1137/15M1035379 apa: Brunner, F., Fischer, J. L., & Knabner, P. (2016). Analysis of a modified second-order mixed hybrid BDM1 finite element method for transport problems in divergence form. Journal on Numerical Analysis. Society for Industrial and Applied Mathematics . https://doi.org/10.1137/15M1035379 chicago: Brunner, Fabian, Julian L Fischer, and Peter Knabner. “Analysis of a Modified Second-Order Mixed Hybrid BDM1 Finite Element Method for Transport Problems in Divergence Form.” Journal on Numerical Analysis. Society for Industrial and Applied Mathematics , 2016. https://doi.org/10.1137/15M1035379. ieee: F. Brunner, J. L. Fischer, and P. Knabner, “Analysis of a modified second-order mixed hybrid BDM1 finite element method for transport problems in divergence form,” Journal on Numerical Analysis, vol. 54, no. 4. Society for Industrial and Applied Mathematics , pp. 2359–2378, 2016. ista: Brunner F, Fischer JL, Knabner P. 2016. Analysis of a modified second-order mixed hybrid BDM1 finite element method for transport problems in divergence form. Journal on Numerical Analysis. 54(4), 2359–2378. mla: Brunner, Fabian, et al. “Analysis of a Modified Second-Order Mixed Hybrid BDM1 Finite Element Method for Transport Problems in Divergence Form.” Journal on Numerical Analysis, vol. 54, no. 4, Society for Industrial and Applied Mathematics , 2016, pp. 2359–78, doi:10.1137/15M1035379. short: F. Brunner, J.L. Fischer, P. Knabner, Journal on Numerical Analysis 54 (2016) 2359–2378. date_created: 2018-12-11T11:51:19Z date_published: 2016-08-02T00:00:00Z date_updated: 2026-05-18T09:48:39Z day: '02' doi: 10.1137/15M1035379 extern: '1' fulldoi: https://doi.org/10.1137/15M1035379 intvolume: ' 54' issue: '4' keyword: - advection-diffusion problem - mixed finite element methods - suboptimal conver-gence - optimal convergence language: - iso: eng month: '08' oa_version: None page: 2359 - 2378 publication: Journal on Numerical Analysis publication_identifier: eissn: - 1095-7170 issnl: - 0036-1429 publication_status: published publisher: 'Society for Industrial and Applied Mathematics ' publist_id: '5954' quality_controlled: '1' status: public title: Analysis of a modified second-order mixed hybrid BDM1 finite element method for transport problems in divergence form type: journal_article user_id: ba8df636-2132-11f1-aed0-ed93e2281fdd volume: 54 year: '2016' ...