---
OA_place: publisher
OA_type: hybrid
_id: '19027'
abstract:
- lang: eng
  text: 'Stochastic PDEs of fluctuating hydrodynamics are a powerful tool for the
    description of fluctuations in many-particle systems. In this paper, we develop
    and analyze a multilevel Monte Carlo (MLMC) scheme for the Dean–Kawasaki equation,
    a pivotal representative of this class of SPDEs. We prove analytically and demonstrate
    numerically that our MLMC scheme provides a significant reduction in computational
    cost (with respect to a standard Monte Carlo method) in the simulation of the
    Dean–Kawasaki equation. Specifically, we link this reduction in cost to having
    a sufficiently large average particle density and show that sizeable cost reductions
    can be obtained even when we have solutions with regions of low density. Numerical
    simulations are provided in the two-dimensional case, confirming our theoretical
    predictions. Our results are formulated entirely in terms of the law of distributions
    rather than in terms of strong spatial norms: this crucially allows for MLMC speed-ups
    altogether despite the Dean–Kawasaki equation being highly singular.'
acknowledgement: The work of the authors was supported by the Austrian Science Fund
  (FWF) projectF65.
article_processing_charge: Yes (in subscription journal)
article_type: original
arxiv: 1
author:
- first_name: Federico
  full_name: Cornalba, Federico
  id: 2CEB641C-A400-11E9-A717-D712E6697425
  last_name: Cornalba
  orcid: 0000-0002-6269-5149
- first_name: Julian L
  full_name: Fischer, Julian L
  id: 2C12A0B0-F248-11E8-B48F-1D18A9856A87
  last_name: Fischer
  orcid: 0000-0002-0479-558X
citation:
  ama: Cornalba F, Fischer JL. Multilevel Monte Carlo methods for the Dean–Kawasaki
    equation from fluctuating hydrodynamics. <i>SIAM Journal on Numerical Analysis</i>.
    2025;63(1):262-287. doi:<a href="https://doi.org/10.1137/23M1617345">10.1137/23M1617345</a>
  apa: Cornalba, F., &#38; Fischer, J. L. (2025). Multilevel Monte Carlo methods for
    the Dean–Kawasaki equation from fluctuating hydrodynamics. <i>SIAM Journal on
    Numerical Analysis</i>. Society for Industrial and Applied Mathematics. <a href="https://doi.org/10.1137/23M1617345">https://doi.org/10.1137/23M1617345</a>
  chicago: Cornalba, Federico, and Julian L Fischer. “Multilevel Monte Carlo Methods
    for the Dean–Kawasaki Equation from Fluctuating Hydrodynamics.” <i>SIAM Journal
    on Numerical Analysis</i>. Society for Industrial and Applied Mathematics, 2025.
    <a href="https://doi.org/10.1137/23M1617345">https://doi.org/10.1137/23M1617345</a>.
  ieee: F. Cornalba and J. L. Fischer, “Multilevel Monte Carlo methods for the Dean–Kawasaki
    equation from fluctuating hydrodynamics,” <i>SIAM Journal on Numerical Analysis</i>,
    vol. 63, no. 1. Society for Industrial and Applied Mathematics, pp. 262–287, 2025.
  ista: Cornalba F, Fischer JL. 2025. Multilevel Monte Carlo methods for the Dean–Kawasaki
    equation from fluctuating hydrodynamics. SIAM Journal on Numerical Analysis. 63(1),
    262–287.
  mla: Cornalba, Federico, and Julian L. Fischer. “Multilevel Monte Carlo Methods
    for the Dean–Kawasaki Equation from Fluctuating Hydrodynamics.” <i>SIAM Journal
    on Numerical Analysis</i>, vol. 63, no. 1, Society for Industrial and Applied
    Mathematics, 2025, pp. 262–87, doi:<a href="https://doi.org/10.1137/23M1617345">10.1137/23M1617345</a>.
  short: F. Cornalba, J.L. Fischer, SIAM Journal on Numerical Analysis 63 (2025) 262–287.
corr_author: '1'
date_created: 2025-02-16T23:02:34Z
date_published: 2025-02-01T00:00:00Z
date_updated: 2025-09-30T10:30:31Z
day: '01'
ddc:
- '510'
department:
- _id: JuFi
doi: 10.1137/23M1617345
external_id:
  arxiv:
  - '2311.08872'
  isi:
  - '001447583400011'
file:
- access_level: open_access
  checksum: 53505647e848ed50f7e0d00c369b14e7
  content_type: application/pdf
  creator: dernst
  date_created: 2025-02-17T08:32:23Z
  date_updated: 2025-02-17T08:32:23Z
  file_id: '19029'
  file_name: 2025_SIAMNumerAnaly_Cornalba.pdf
  file_size: 2435019
  relation: main_file
  success: 1
file_date_updated: 2025-02-17T08:32:23Z
has_accepted_license: '1'
intvolume: '        63'
isi: 1
issue: '1'
language:
- iso: eng
license: https://creativecommons.org/licenses/by/4.0/
month: '02'
oa: 1
oa_version: Published Version
page: 262-287
project:
- _id: fc31cba2-9c52-11eb-aca3-ff467d239cd2
  grant_number: F6504
  name: Taming Complexity in Partial Differential Systems
publication: SIAM Journal on Numerical Analysis
publication_identifier:
  eissn:
  - 1095-7170
  issn:
  - 0036-1429
publication_status: published
publisher: Society for Industrial and Applied Mathematics
quality_controlled: '1'
scopus_import: '1'
status: public
title: Multilevel Monte Carlo methods for the Dean–Kawasaki equation from fluctuating
  hydrodynamics
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: 63
year: '2025'
...
---
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. <i>Journal
    on Numerical Analysis</i>. 2016;54(4):2359-2378. doi:<a href="https://doi.org/10.1137/15M1035379">10.1137/15M1035379</a>
  apa: Brunner, F., Fischer, J. L., &#38; Knabner, P. (2016). Analysis of a modified
    second-order mixed hybrid BDM1 finite element method for transport problems in
    divergence form. <i>Journal on Numerical Analysis</i>. Society for Industrial
    and Applied Mathematics . <a href="https://doi.org/10.1137/15M1035379">https://doi.org/10.1137/15M1035379</a>
  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.” <i>Journal on Numerical Analysis</i>. Society for Industrial
    and Applied Mathematics , 2016. <a href="https://doi.org/10.1137/15M1035379">https://doi.org/10.1137/15M1035379</a>.
  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,”
    <i>Journal on Numerical Analysis</i>, 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.” <i>Journal on
    Numerical Analysis</i>, vol. 54, no. 4, Society for Industrial and Applied Mathematics
    , 2016, pp. 2359–78, doi:<a href="https://doi.org/10.1137/15M1035379">10.1137/15M1035379</a>.
  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'
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'
...