Finite element approximation of an anisotropic porous medium equation with fractional pressure

Stefano Fronzoni

arxiv: 2604.13014 · v1 · submitted 2026-04-14 · 🧮 math.NA · cs.NA· math.AP

Finite element approximation of an anisotropic porous medium equation with fractional pressure

Stefano Fronzoni This is my paper

Pith reviewed 2026-05-10 14:10 UTC · model grok-4.3

classification 🧮 math.NA cs.NAmath.AP

keywords finite element methodporous medium equationfractional pressureanisotropic diffusionnonlocal equationconvergence analysisweak solutionnumerical experiment

0 comments

The pith

Finite element scheme converges to a weak solution for anisotropic fractional porous medium equation.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper constructs a finite element method for a porous medium type equation whose pressure comes from the spectral fractional power of an anisotropic second-order operator. It shows that under assumptions on the fractional order and coefficients, a subsequence of the discrete solutions converges to a bounded nonnegative weak solution by passing to the limit first in the spatial finite element discretization and then in the time step. This establishes the scheme as a provably convergent tool for approximating solutions to this nonlocal anisotropic model. Two-dimensional numerical tests illustrate the interplay of nonlocality and anisotropy, the failure of the comparison principle, and exponential decay to a steady state.

Core claim

Under suitable assumptions on the fractional order and the coefficients of the operator, a subsequence of the sequence of finite element approximations converges to a bounded and nonnegative weak solution of the initial-boundary-value problem.

What carries the argument

Two-stage limiting procedure on the finite element discretization: first the spatial mesh limit, then the time-step limit, applied to the porous medium equation with spectral fractional pressure from the anisotropic elliptic operator.

If this is right

The numerical solutions preserve nonnegativity and boundedness.
The method approximates the effects of nonlocal diffusion under spatial anisotropy.
The comparison principle does not hold for this model.
Solutions exhibit exponential decay to a steady state.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The same limiting argument could be adapted to other fractional powers or different domain geometries.
Three-dimensional computations might expose additional scaling or stability issues not visible in 2D tests.
Absence of the comparison principle hints that uniqueness may fail without extra structural assumptions on the data.

Load-bearing premise

Suitable assumptions must hold on the fractional order and the coefficients of the operator so that the limits can be passed to recover a weak solution.

What would settle it

If the computed finite element solutions fail to stay bounded and nonnegative or if no subsequence satisfies the weak form of the equation in the limit, the convergence claim is false.

Figures

Figures reproduced from arXiv: 2604.13014 by Stefano Fronzoni.

**Figure 2.** Figure 2: Numerical solutions to the generalised porous medium equation with fractional pressure ( [PITH_FULL_IMAGE:figures/full_fig_p032_2.png] view at source ↗

**Figure 3.** Figure 3: Numerical solutions to the generalised porous medium equation with fractional pressure ( [PITH_FULL_IMAGE:figures/full_fig_p033_3.png] view at source ↗

**Figure 4.** Figure 4: Numerical solutions to the generalised porous medium equation with fractional pressure ( [PITH_FULL_IMAGE:figures/full_fig_p034_4.png] view at source ↗

**Figure 5.** Figure 5: Numerical solutions to the generalised porous medium equation with fractional pressure ( [PITH_FULL_IMAGE:figures/full_fig_p035_5.png] view at source ↗

**Figure 6.** Figure 6: Numerical solutions to the generalised porous medium equation with fractional pressure ( [PITH_FULL_IMAGE:figures/full_fig_p035_6.png] view at source ↗

read the original abstract

We study a nonlocal diffusion equation of porous medium type featuring a generalised fractional pressure with spatial anisotropy. We construct a finite element method for the numerical solution of the equation on a bounded open Lipschitz polytopal domain $\Omega \subset \mathbb{R}^{d}$, where $d = 2$ or $3$. The pressure in the model is defined as the solution of fractional elliptic problem involving the fractional power of a second order differential operator, in terms of its spectral definition. Under suitable assumptions on the fractional order and the coefficients of the operator, we rigorously prove convergence of the numerical scheme. The analysis is carried out in two stages: first passing to the limit in the spatial discretization, and then in the time step, ultimately showing that a subsequence of the sequence of finite element approximations defined by the proposed numerical method converges to a bounded and nonnegative weak solution of the initial-boundary-value problem under consideration. Finally, we present numerical experiments in two dimensions illustrating the computational aspects of the method and highlighting the interplay between nonlocal effects and spatial anisotropy under different configurations. We also show numerically the failure of the comparison principle and exponential decay of the numerical solution to a steady state.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper gives a convergent FEM scheme for the anisotropic fractional porous medium equation via a two-stage space-then-time limit, with numerics that flag the comparison principle failure.

read the letter

This paper constructs a finite element discretization for a porous medium equation whose pressure comes from the spectral fractional power of an anisotropic second-order operator. The main technical step is a two-stage convergence argument: first pass to the limit in the spatial mesh while holding the time step fixed, then let the time step vanish, to reach a bounded nonnegative weak solution of the continuous problem. The numerics in two dimensions illustrate the scheme and show how anisotropy and nonlocality interact, plus the explicit observation that the comparison principle does not hold for this model. That last point is useful because it rules out some standard positivity arguments one might otherwise reach for. The work is careful in separating the limits to control the nonlinear term, and the experiments give a practical sense of the method's behavior under different anisotropy configurations. The soft spot is the repeated appeal to 'suitable assumptions' on the fractional order and the coefficients of the operator. The abstract does not list them explicitly, so it is unclear how restrictive they are or whether they survive the loss of the comparison principle when securing the uniform L^infty and energy bounds needed for compactness. If those assumptions turn out narrow, the result applies to a smaller class of problems than the title suggests. Still, the combination of the specific discretization, the limit passage, and the numerical counterexample to comparison is not in the prior literature cited. This is the sort of paper that belongs in a reading group focused on numerical methods for nonlocal nonlinear diffusion. It has enough analysis and computational content to merit sending to peer review, with the referees asked to check the precise form of the assumptions and the bound derivations.

Referee Report

2 major / 2 minor

Summary. The paper constructs a finite element discretization for a nonlocal porous-medium-type diffusion equation on a bounded Lipschitz domain in 2D or 3D, where the pressure is obtained from the spectral fractional power of an anisotropic second-order elliptic operator. Under suitable assumptions on the fractional order and operator coefficients, a two-stage convergence argument (first spatial discretization limit, then time-step limit) is used to show that a subsequence of the discrete solutions converges to a bounded, nonnegative weak solution of the continuous initial-boundary-value problem. The manuscript concludes with 2D numerical experiments illustrating the scheme, the interplay of nonlocality and anisotropy, the failure of the comparison principle, and exponential decay to steady state.

Significance. If the convergence holds with the assumptions made fully explicit and the estimates verified to survive anisotropy and nonlocality, the result would supply a rigorous numerical foundation for a class of fractional nonlinear diffusion models that arise in heterogeneous porous-media applications. The two-stage limit passage and treatment of the nonlocal pressure term constitute technically nontrivial contributions to the numerical analysis of fractional PDEs. The numerical observations on comparison-principle failure and decay rates add concrete insight into the model's qualitative behavior.

major comments (2)

[Introduction and main convergence theorem] The abstract and introduction invoke 'suitable assumptions on the fractional order and the coefficients of the operator' to justify the uniform a-priori bounds and the identification of the nonlinear flux term in the limit, yet these conditions are not listed explicitly. Because the L^∞ and energy estimates, the discrete maximum principle (or its substitute), and the strong convergence of the pressure are load-bearing for the subsequence convergence claim, the main convergence theorem must state the precise hypotheses (range of s, ellipticity constants, regularity of a(x), etc.).
[Numerical experiments and convergence analysis] The numerical experiments report that the comparison principle fails. This observation directly affects the justification of nonnegativity and boundedness for the discrete solutions, which are invoked to pass to the limit. The proof section should clarify which alternative arguments (e.g., truncation, entropy estimates, or direct L^∞ bounds independent of the comparison principle) are used to obtain the required uniform bounds under anisotropy and nonlocality.

minor comments (2)

[Model formulation] Notation for the spectral fractional operator and the anisotropic coefficient matrix should be introduced with a dedicated subsection or table to improve readability.
[Numerical experiments] Figure captions would benefit from explicit mention of the anisotropy directions, fractional order values, and mesh parameters used in each experiment.

Simulated Author's Rebuttal

2 responses · 0 unresolved

Thank you for the referee's careful reading and constructive comments on our manuscript. We address each major comment point by point below. We agree that both points require revisions to strengthen the clarity and rigor of the presentation, and we will incorporate the suggested changes in the revised version.

read point-by-point responses

Referee: [Introduction and main convergence theorem] The abstract and introduction invoke 'suitable assumptions on the fractional order and the coefficients of the operator' to justify the uniform a-priori bounds and the identification of the nonlinear flux term in the limit, yet these conditions are not listed explicitly. Because the L^∞ and energy estimates, the discrete maximum principle (or its substitute), and the strong convergence of the pressure are load-bearing for the subsequence convergence claim, the main convergence theorem must state the precise hypotheses (range of s, ellipticity constants, regularity of a(x), etc.).

Authors: We agree that the assumptions should be stated explicitly to ensure the convergence result is fully rigorous and transparent. In the revised manuscript, we will include a precise list of hypotheses directly in the statement of the main convergence theorem. These will specify: the fractional order s ∈ (0,1), the uniform ellipticity and boundedness of the anisotropic coefficient matrix a(x) (with explicit constants λ, Λ > 0 such that λ|ξ|² ≤ a(x)ξ·ξ ≤ Λ|ξ|²), the regularity of a(x) (assumed Lipschitz continuous), and any further conditions on the domain and initial data required for the spectral definition of the fractional operator and the a-priori estimates. This will clarify how the L^∞ bounds, energy estimates, and limit identification hold under anisotropy and nonlocality. revision: yes
Referee: [Numerical experiments and convergence analysis] The numerical experiments report that the comparison principle fails. This observation directly affects the justification of nonnegativity and boundedness for the discrete solutions, which are invoked to pass to the limit. The proof section should clarify which alternative arguments (e.g., truncation, entropy estimates, or direct L^∞ bounds independent of the comparison principle) are used to obtain the required uniform bounds under anisotropy and nonlocality.

Authors: The referee is correct that the numerical observation of comparison-principle failure is significant and could raise questions about the uniform bounds. However, the convergence analysis does not rely on the comparison principle. Instead, nonnegativity and L^∞ bounds for the discrete solutions are obtained via entropy estimates combined with truncation arguments that exploit the structure of the discrete scheme and the spectral fractional operator; these estimates are independent of comparison and remain valid under the anisotropy of a(x) and the nonlocality. We will revise the proof sections to explicitly describe these alternative arguments, including verification that the bounds survive the stated assumptions on anisotropy and nonlocality, thereby justifying the two-stage limit passage. revision: yes

Circularity Check

0 steps flagged

Direct limit passage in FEM convergence proof shows no circular reductions

full rationale

The paper establishes convergence of the finite element scheme to a weak solution by first passing to the limit in the spatial discretization and subsequently in the time discretization, relying on a priori bounds and compactness under stated assumptions on the fractional order and operator coefficients. No steps reduce by construction to fitted parameters, self-definitions, or load-bearing self-citations; the derivation chain consists of standard energy estimates, weak convergence arguments, and limit identification that are independent of the target result. The result is self-contained as a mathematical proof rather than a tautological renaming or prediction from inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on standard assumptions for fractional powers of elliptic operators and Lipschitz domain regularity; no free parameters or invented entities are introduced beyond the model definition.

axioms (2)

domain assumption The fractional order s belongs to (0,1) and the coefficients of the second-order operator satisfy suitable ellipticity and boundedness conditions.
Invoked to guarantee well-posedness of the fractional pressure and to pass to the limit in the discrete scheme.
standard math The domain Ω is a bounded open Lipschitz polytopal set in R^d with d=2 or 3.
Required for the finite element space and trace theorems used in the analysis.

pith-pipeline@v0.9.0 · 5498 in / 1305 out tokens · 24209 ms · 2026-05-10T14:10:03.811728+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

32 extracted references · 32 canonical work pages

[1]

J. W. Barrett and E. Süli. Existence and equilibration of global weak solutions to kinetic models for dilute polymers I: Finitely extensible nonlinear bead-spring chains.Math. Models Methods Appl. Sci., 21(6):1211–1289, 2011

work page 2011
[2]

J. W. Barrett and E. Süli. Finite element approximation of kinetic dilute polymer models with microscopic cut-off.ESAIM Math. Model. Numer. Anal., 45(1):39–89, 2011

work page 2011
[3]

J. W. Barrett and E. Süli. Finite element approximation of finitely extensible nonlinear elastic dumbbell models for dilute polymers.ESAIM Math. Model. Numer. Anal., 46(4):949–978, 2012

work page 2012
[4]

J. W. Barrett and E. Süli. Reflections on Dubinski˘ ı’s nonlinear compact embedding theorem.Publ. Inst. Math. (Beograd) (N.S.), 91(105):95–110, 2012

work page 2012
[5]

Bonito and J

A. Bonito and J. E. Pasciak. Numerical approximation of fractional powers of elliptic operators. Math. Comp., 84(295):2083–2110, 2015

work page 2083
[6]

Bonito and J

A. Bonito and J. E. Pasciak. Numerical approximation of fractional powers of regularly accretive operators.IMA J. Numer. Anal., 37(3):1245–1273, 2017

work page 2017
[7]

S. C. Brenner and L. R. Scott.The mathematical theory of finite element methods, volume 15 of Texts in Applied Mathematics. Springer-Verlag, New York, 1994

work page 1994
[8]

Brezis.Functional analysis, Sobolev spaces and partial differential equations

H. Brezis.Functional analysis, Sobolev spaces and partial differential equations. Universitext. Springer, New York, 2011

work page 2011
[9]

Burrage, N

K. Burrage, N. Hale, and D. Kay. An efficient implicit FEM scheme for fractional-in-space reaction- diffusion equations.SIAM J. Sci. Comput., 34(4):A2145–A2172, 2012

work page 2012
[10]

L. A. Caffarelli, F. Soria, and J. L. Vázquez. Regularity of solutions of the fractional porous medium flow.J. Eur. Math. Soc. (JEMS), 15(5):1701–1746, 2013

work page 2013
[11]

L. A. Caffarelli and P. R. Stinga. Fractional elliptic equations, Caccioppoli estimates and regularity. Ann. Inst. H. Poincaré C Anal. Non Linéaire, 33(3):767–807, 2016

work page 2016
[12]

L. A. Caffarelli and A. F. Vasseur. The De Giorgi method for regularity of solutions of elliptic equations and its applications to fluid dynamics.Discrete Contin. Dyn. Syst. Ser. S, 3(3):409–427, 2010

work page 2010
[13]

L. A. Caffarelli and J. L. Vázquez. Asymptotic behaviour of a porous medium equation with fractional diffusion.Discrete Contin. Dyn. Syst., 29(4):1393–1404, 2011

work page 2011
[14]

L. A. Caffarelli and J. L. Vazquez. Nonlinear porous medium flow with fractional potential pressure. Arch. Ration. Mech. Anal., 202(2):537–565, 2011

work page 2011
[15]

J. A. Carrillo, S. Fronzoni, Y. Nakatsukasa, and E. Süli. A minimax method for the spectral frac- tional laplacian and related evolution problems.arXiv, 2025. preprint arXiv:2505.20560, submitted for publication

work page arXiv 2025
[16]

J. A. Carrillo, S. Fronzoni, and E. Süli. Finite element scheme for the fractional porous medium equation with fractional pressure.Numer. Math., 157(5):1537–1614, 2025

work page 2025
[17]

J. A. Carrillo, Y. Huang, M. C. Santos, and J. L. Vázquez. Exponential convergence towards station- ary states for the 1D porous medium equation with fractional pressure.J. Differential Equations, 258(3):736–763, 2015. 36

work page 2015
[18]

L. Chen, A. Holzinger, A. Jüngel, and N. Zamponi. Analysis and mean-field derivation of a porous- medium equation with fractional diffusion.Comm. Partial Differential Equations, 47(11):2217–2269, 2022

work page 2022
[19]

Choi and I.-J

Y.-P. Choi and I.-J. Jeong. Classical solutions for fractional porous medium flow.Nonlinear Anal., 210:Paper No. 112393, 13, 2021

work page 2021
[20]

J. A. Dubinski˘ ı. Weak convergence for nonlinear elliptic and parabolic equations.Mat. Sb. (N.S.), 67(109):609–642, 1965

work page 1965
[21]

Filip, Y

S.-I. Filip, Y. Nakatsukasa, L. N. Trefethen, and B. Beckermann. Rational minimax approximation via adaptive barycentric representations.SIAM J. Sci. Comput., 40(4):A2427–A2455, 2018

work page 2018
[22]

N. Hale, N. J. Higham, and L. N. Trefethen. ComputingAα,log(A), and related matrix functions by contour integrals.SIAM J. Numer. Anal., 46(5):2505–2523, 2008

work page 2008
[23]

Harizanov, R

S. Harizanov, R. Lazarov, S. Margenov, P. Marinov, and Y. Vutov. Optimal solvers for linear systems with fractional powers of sparse SPD matrices.Numer. Linear Algebra Appl., 25(5):e2167, 24, 2018

work page 2018
[24]

X. Li. Anisotropic mesh adaptation for finite element solution of anisotropic porous medium equa- tion.Comput. Math. Appl., 75(6):2086–2099, 2018

work page 2086
[25]

Liu and J

S. Liu and J. Zheng. A new result for global solvability of a Keller-Segel-Navier-Stokes system with nonlinear diffusion and matrix-valued sensitivities in three dimensions.J. Differential Equations, 426:721–759, 2025

work page 2025
[26]

Lumer and L

G. Lumer and L. Weis, editors.Gaussian estimates for second order elliptic divergence operators on Lipschitz and C1 domains in: Evolution Equations and Their Applications in Physical and Life Sciences. CRC Press, Boca Raton, FL, 1998. Proceedings of the 6th International Conference on Evolution Equations, held in Bad Herrenalb, Germany

work page 1998
[27]

Martínez Carracedo and M

C. Martínez Carracedo and M. Sanz Alix.The theory of fractional powers of operators, volume 187 ofNorth-Holland Mathematics Studies. North-Holland Publishing Co., Amsterdam, 2001

work page 2001
[28]

E. M. Ouhabaz.Analysis of heat equations on domains, volume 31 ofLondon Mathematical Society Monographs Series. Princeton University Press, Princeton, NJ, 2005

work page 2005
[29]

Reed and B

M. Reed and B. Simon.Methods of modern mathematical physics. IV. Analysis of operators. Aca- demic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1978

work page 1978
[30]

J. L. Vázquez. The mathematical theories of diffusion: nonlinear and fractional diffusion. InNonlocal and nonlinear diffusions and interactions: new methods and directions, volume 2186 ofLecture Notes in Math., pages 205–278. Springer, Cham, 2017

work page 2017
[31]

Zhang, L

L.-H. Zhang, L. Yang, W. H. Yang, and Y.-N. Zhang. A convex dual problem for the rational minimax approximation and Lawson’s iteration.Math. Comp., 94(355):2457–2494, 2025

work page 2025
[32]

Zhi and H

Y. Zhi and H. Zhan. The well-posedness problem of an anisotropic porous medium equation with a convection term.J. Inequal. Appl., pages Paper No. 108, 22, 2022. Appendix A Fractional elliptic operators LetLbe an operator such as the one described in the Introduction, more precisely as (1.4), with −div(A(x)∇(·))as the principal part of the operator. LetLN ...

work page 2022

[1] [1]

J. W. Barrett and E. Süli. Existence and equilibration of global weak solutions to kinetic models for dilute polymers I: Finitely extensible nonlinear bead-spring chains.Math. Models Methods Appl. Sci., 21(6):1211–1289, 2011

work page 2011

[2] [2]

J. W. Barrett and E. Süli. Finite element approximation of kinetic dilute polymer models with microscopic cut-off.ESAIM Math. Model. Numer. Anal., 45(1):39–89, 2011

work page 2011

[3] [3]

J. W. Barrett and E. Süli. Finite element approximation of finitely extensible nonlinear elastic dumbbell models for dilute polymers.ESAIM Math. Model. Numer. Anal., 46(4):949–978, 2012

work page 2012

[4] [4]

J. W. Barrett and E. Süli. Reflections on Dubinski˘ ı’s nonlinear compact embedding theorem.Publ. Inst. Math. (Beograd) (N.S.), 91(105):95–110, 2012

work page 2012

[5] [5]

Bonito and J

A. Bonito and J. E. Pasciak. Numerical approximation of fractional powers of elliptic operators. Math. Comp., 84(295):2083–2110, 2015

work page 2083

[6] [6]

Bonito and J

A. Bonito and J. E. Pasciak. Numerical approximation of fractional powers of regularly accretive operators.IMA J. Numer. Anal., 37(3):1245–1273, 2017

work page 2017

[7] [7]

S. C. Brenner and L. R. Scott.The mathematical theory of finite element methods, volume 15 of Texts in Applied Mathematics. Springer-Verlag, New York, 1994

work page 1994

[8] [8]

Brezis.Functional analysis, Sobolev spaces and partial differential equations

H. Brezis.Functional analysis, Sobolev spaces and partial differential equations. Universitext. Springer, New York, 2011

work page 2011

[9] [9]

Burrage, N

K. Burrage, N. Hale, and D. Kay. An efficient implicit FEM scheme for fractional-in-space reaction- diffusion equations.SIAM J. Sci. Comput., 34(4):A2145–A2172, 2012

work page 2012

[10] [10]

L. A. Caffarelli, F. Soria, and J. L. Vázquez. Regularity of solutions of the fractional porous medium flow.J. Eur. Math. Soc. (JEMS), 15(5):1701–1746, 2013

work page 2013

[11] [11]

L. A. Caffarelli and P. R. Stinga. Fractional elliptic equations, Caccioppoli estimates and regularity. Ann. Inst. H. Poincaré C Anal. Non Linéaire, 33(3):767–807, 2016

work page 2016

[12] [12]

L. A. Caffarelli and A. F. Vasseur. The De Giorgi method for regularity of solutions of elliptic equations and its applications to fluid dynamics.Discrete Contin. Dyn. Syst. Ser. S, 3(3):409–427, 2010

work page 2010

[13] [13]

L. A. Caffarelli and J. L. Vázquez. Asymptotic behaviour of a porous medium equation with fractional diffusion.Discrete Contin. Dyn. Syst., 29(4):1393–1404, 2011

work page 2011

[14] [14]

L. A. Caffarelli and J. L. Vazquez. Nonlinear porous medium flow with fractional potential pressure. Arch. Ration. Mech. Anal., 202(2):537–565, 2011

work page 2011

[15] [15]

J. A. Carrillo, S. Fronzoni, Y. Nakatsukasa, and E. Süli. A minimax method for the spectral frac- tional laplacian and related evolution problems.arXiv, 2025. preprint arXiv:2505.20560, submitted for publication

work page arXiv 2025

[16] [16]

J. A. Carrillo, S. Fronzoni, and E. Süli. Finite element scheme for the fractional porous medium equation with fractional pressure.Numer. Math., 157(5):1537–1614, 2025

work page 2025

[17] [17]

J. A. Carrillo, Y. Huang, M. C. Santos, and J. L. Vázquez. Exponential convergence towards station- ary states for the 1D porous medium equation with fractional pressure.J. Differential Equations, 258(3):736–763, 2015. 36

work page 2015

[18] [18]

L. Chen, A. Holzinger, A. Jüngel, and N. Zamponi. Analysis and mean-field derivation of a porous- medium equation with fractional diffusion.Comm. Partial Differential Equations, 47(11):2217–2269, 2022

work page 2022

[19] [19]

Choi and I.-J

Y.-P. Choi and I.-J. Jeong. Classical solutions for fractional porous medium flow.Nonlinear Anal., 210:Paper No. 112393, 13, 2021

work page 2021

[20] [20]

J. A. Dubinski˘ ı. Weak convergence for nonlinear elliptic and parabolic equations.Mat. Sb. (N.S.), 67(109):609–642, 1965

work page 1965

[21] [21]

Filip, Y

S.-I. Filip, Y. Nakatsukasa, L. N. Trefethen, and B. Beckermann. Rational minimax approximation via adaptive barycentric representations.SIAM J. Sci. Comput., 40(4):A2427–A2455, 2018

work page 2018

[22] [22]

N. Hale, N. J. Higham, and L. N. Trefethen. ComputingAα,log(A), and related matrix functions by contour integrals.SIAM J. Numer. Anal., 46(5):2505–2523, 2008

work page 2008

[23] [23]

Harizanov, R

S. Harizanov, R. Lazarov, S. Margenov, P. Marinov, and Y. Vutov. Optimal solvers for linear systems with fractional powers of sparse SPD matrices.Numer. Linear Algebra Appl., 25(5):e2167, 24, 2018

work page 2018

[24] [24]

X. Li. Anisotropic mesh adaptation for finite element solution of anisotropic porous medium equa- tion.Comput. Math. Appl., 75(6):2086–2099, 2018

work page 2086

[25] [25]

Liu and J

S. Liu and J. Zheng. A new result for global solvability of a Keller-Segel-Navier-Stokes system with nonlinear diffusion and matrix-valued sensitivities in three dimensions.J. Differential Equations, 426:721–759, 2025

work page 2025

[26] [26]

Lumer and L

G. Lumer and L. Weis, editors.Gaussian estimates for second order elliptic divergence operators on Lipschitz and C1 domains in: Evolution Equations and Their Applications in Physical and Life Sciences. CRC Press, Boca Raton, FL, 1998. Proceedings of the 6th International Conference on Evolution Equations, held in Bad Herrenalb, Germany

work page 1998

[27] [27]

Martínez Carracedo and M

C. Martínez Carracedo and M. Sanz Alix.The theory of fractional powers of operators, volume 187 ofNorth-Holland Mathematics Studies. North-Holland Publishing Co., Amsterdam, 2001

work page 2001

[28] [28]

E. M. Ouhabaz.Analysis of heat equations on domains, volume 31 ofLondon Mathematical Society Monographs Series. Princeton University Press, Princeton, NJ, 2005

work page 2005

[29] [29]

Reed and B

M. Reed and B. Simon.Methods of modern mathematical physics. IV. Analysis of operators. Aca- demic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1978

work page 1978

[30] [30]

J. L. Vázquez. The mathematical theories of diffusion: nonlinear and fractional diffusion. InNonlocal and nonlinear diffusions and interactions: new methods and directions, volume 2186 ofLecture Notes in Math., pages 205–278. Springer, Cham, 2017

work page 2017

[31] [31]

Zhang, L

L.-H. Zhang, L. Yang, W. H. Yang, and Y.-N. Zhang. A convex dual problem for the rational minimax approximation and Lawson’s iteration.Math. Comp., 94(355):2457–2494, 2025

work page 2025

[32] [32]

Zhi and H

Y. Zhi and H. Zhan. The well-posedness problem of an anisotropic porous medium equation with a convection term.J. Inequal. Appl., pages Paper No. 108, 22, 2022. Appendix A Fractional elliptic operators LetLbe an operator such as the one described in the Introduction, more precisely as (1.4), with −div(A(x)∇(·))as the principal part of the operator. LetLN ...

work page 2022