@article{18706,
  abstract     = {We prove discrete-to-continuum convergence for dynamical optimal transport on  Zd
 -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.},
  author       = {Portinale, Lorenzo and Quattrocchi, Filippo},
  issn         = {1469-4425},
  journal      = {European Journal of Applied Mathematics},
  keywords     = {optimal transport, discrete-to-continuum, homogenisation, linear growth, gamma-convergence},
  number       = {3},
  pages        = {614--642},
  publisher    = {Cambridge University Press},
  title        = {{Discrete-to-continuum limits of optimal transport with linear growth on periodic graphs}},
  doi          = {10.1017/s0956792524000810},
  volume       = {37},
  year         = {2026},
}

@article{11739,
  abstract     = {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.},
  author       = {Forkert, Dominik L and Maas, Jan and Portinale, Lorenzo},
  issn         = {1095-7154},
  journal      = {SIAM Journal on Mathematical Analysis},
  keywords     = {Fokker--Planck equation, gradient flow, evolutionary $\Gamma$-convergence},
  number       = {4},
  pages        = {4297--4333},
  publisher    = {Society for Industrial and Applied Mathematics},
  title        = {{Evolutionary $\Gamma$-convergence of entropic gradient flow structures for Fokker-Planck equations in multiple dimensions}},
  doi          = {10.1137/21M1410968},
  volume       = {54},
  year         = {2022},
}

@article{1315,
  abstract     = {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.},
  author       = {Brunner, Fabian and Fischer, Julian L and Knabner, Peter},
  issn         = {1095-7170},
  journal      = {Journal on Numerical Analysis},
  keywords     = {advection-diffusion problem, mixed finite element methods, suboptimal conver-gence, optimal convergence},
  number       = {4},
  pages        = {2359 -- 2378},
  publisher    = {Society for Industrial and Applied Mathematics },
  title        = {{Analysis of a modified second-order mixed hybrid BDM1 finite element method for transport problems in divergence form}},
  doi          = {10.1137/15M1035379},
  volume       = {54},
  year         = {2016},
}