Carlo Marcati


Last Name

Marcati

First Name

Carlo

ORCID

0000-0002-0313-5387

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2019 - 201932020 - 202626
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Publications1 - 10 of 29
  • Exponential Convergence of hp-FEM for the Integral Fractional Laplacian on cuboids
    Item type: Report
    Bahr, Björn; Faustmann, Markus; Marcati, Carlo; et al. (2026)
    SAM Research Report
    For the Dirichlet integral fractional Laplacian, we prove root exponential convergence of tensor-product ℎ-finite element approximations on (01)3, for forcing that is analytic in [01]3. Exploiting analytic regularity estimates in weighted Sobolev spaces [10], we prove for ℎ-GLLinterpolation approximations with degrees of free domthe energy norm error bound ≲ exp(− 6√ ). Tensor product mesh families whichare geometrically refined towards all sides of (01)3 are used. Numerical experiments with ℎ-Galerkin FEM confirm the bound.
  • Weighted analytic regularity for the integral fractional Laplacian in polyhedra
    Item type: Report
    Faustmann, Markus; Marcati, Carlo; Melenk, Jens Markus; et al. (2023)
    SAM Research Report
    On polytopal domains in 3D, we prove weighted analytic regularity of solutions to the Dirichlet problem for the integral fractional Laplacian with analytic right-hand side. Employing the Caffarelli-Silvestre extension allows to localize the problem and to decompose the regularity estimates into results on vertex, edge, face, vertex-edge, vertex-face, edge-face and vertex-edge-face neighborhoods of the boundary. Using tangential differentiability of the extended solutions, a bootstrapping argument based on Caccioppoli inequalities on dyadic decompositions of the neighborhoods provides control of higher order derivatives.
  • Weighted Analytic Regularity for the Integral Fractional Laplacian in Polyhedra
    Item type: Journal Article
    Faustmann, Markus; Marcati, Carlo; Melenk, Jens Markus; et al. (2025)
    Analysis and Applications
    On polytopal domains in $\mathbb{R}^3$ we prove weighted analytic regularity of solutions to the Dirichlet problem for the shifted integral fractional Laplacian with analytic right-hand side. Employing the Caffarelli–Silvestre extension allows one to localize the problem and to decompose the regularity estimates into results on vertex, edge, face, vertex-edge, vertex-face, edge-face and vertex-edge-face neighborhoods of the boundary. Using tangential differentiability of the extended solutions, a bootstrapping argument based on Caccioppoli inequalities on dyadic decompositions of the neighborhoods provides weighted, analytic control of higher-order solution derivatives.
  • Weighted analyticity of Hartree-Fock eigenfunctions
    Item type: Report
    Maday, Yvon; Marcati, Carlo (2020)
    SAM Research Report
    We prove analytic-type estimates in weighted Sobolev spaces on the eigenfunctions of a class of elliptic and nonlinear eigenvalue problems with singular potentials, which includes the Hartree-Fock equations. Going beyond classical results on the analyticity of the wavefunctions away from the nuclei, we prove weighted estimates locally at each singular point, with precise control of the derivatives of all orders. Our estimates have far-reaching consequences for the approximation of the eigenfunctions of the problems considered, and they can be used to prove a priori estimates on the numerical solution of such eigenvalue problems.
  • Exponential ReLU Neural Network Approximation Rates for Point and Edge Singularities
    Item type: Report
    Marcati, Carlo; Opschoor, Joost A.A.; Petersen, Philipp C.; et al. (2020)
    SAM Research Report
    We prove exponential expressivity with stable ReLU Neural Networks (ReLU NNs) in H1(Ω) for weighted analytic function classes in certain polytopal domains Ω, in space dimension d=2,3. Functions in these classes are locally analytic on open subdomains D⊂Ω, but may exhibit isolated point singularities in the interior of Ω or corner and edge singularities at the boundary ∂Ω. The exponential expression rate bounds proved here imply uniform exponential expressivity by ReLU NNs of solution families for several elliptic boundary and eigenvalue problems with analytic data. The exponential approximation rates are shown to hold in space dimension d=2 on Lipschitz polygons with straight sides, and in space dimension d=3 on Fichera-type polyhedral domains with plane faces. The constructive proofs indicate in particular that NN depth and size increase poly-logarithmically with respect to the target NN approximation accuracy ε>0 in H1(Ω). The results cover in particular solution sets of linear, second order elliptic PDEs with analytic data and certain nonlinear elliptic eigenvalue problems with analytic nonlinearities and singular, weighted analytic potentials as arise in electron structure models. In the latter case, the functions correspond to electron densities that exhibit isolated point singularities at the positions of the nuclei. Our findings provide in particular mathematical foundation of recently reported, successful uses of deep neural networks in variational electron structure algorithms.
  • Exponential Convergence of hp-FEM for the Integral Fractional Laplacian in 1D
    Item type: Conference Paper
    Bahr, Björn; Faustmann, Markus; Marcati, Carlo; et al. (2023)
    Lecture Notes in Computational Science and Engineering ~ Spectral and High Order Methods for Partial Differential Equations ICOSAHOM 2020+1
    We prove weighted analytic regularity for the solution of the integral fractional Poisson problem on bounded intervals with analytic right-hand side. Based on this regularity result, we prove exponential convergence of the hp-FEM on geometric boundary-refined meshes.
  • Analytic Regularity for the Incompressible Navier-Stokes Equations in Polygons
    Item type: Journal Article
    Marcati, Carlo; Schwab, Christoph (2020)
    SIAM Journal on Mathematical Analysis
    In a plane polygon P with straight sides, we prove analytic regularity of the Leray-Hopf solution of the stationary, viscous, and incompressible Navier-Stokes equations. We assume small data, analytic volume force, and no-slip boundary conditions. Analytic regularity is quantified in so-called countably normed, corner-weighted spaces with homogeneous norms. Implications of this analytic regularity include exponential smallness of Kolmogorov N-widths of solutions, exponential convergence rates of mixed hp-discontinuous Galerkin finite element and spectral element discretizations, and model order reduction techniques. © 2020 Society for Industrial and Applied Mathematics.
  • Exponential convergence of mixed hp-DGFEM for the incompressible Navier-Stokes equations in R²
    Item type: Report
    Schötzau, Dominik; Marcati, Carlo; Schwab, Christoph (2020)
    SAM Research Report
    In a polygon Ω ⊂ R2, we consider mixed hp-discontinuous Galerkin approximations of the stationary, incompressible Navier-Stokes equations, subject to no-slip boundary conditions. We use geometrically corner-refined meshes and hp spaces with linearly increasing polynomial degrees. Based on recent results on analytic regularity of velocity field and pressure of Leray solutions in Ω, we prove exponential rates of convergence of the mixed hp-discontinuous Galerkin finite element method (hp-DGFEM), with respect to the number of degrees of freedom, for small data which is piecewise analytic
  • p- and hp- virtual elements for the Stokes problem
    Item type: Journal Article
    Chernov, Alexey; Marcati, Carlo; Mascotto, Lorenzo (2021)
    Advances in Computational Mathematics
    We analyse the p- and hp-versions of the virtual element method (VEM) for the Stokes problem on polygonal domains. The key tool in the analysis is the existence of a bijection between Poisson-like and Stokes-like VE spaces for the velocities. This allows us to re-interpret the standard VEM for Stokes as a VEM, where the test and trial discrete velocities are sought in Poisson-like VE spaces. The upside of this fact is that we inherit from Beirão da Veiga et al. (Numer. Math. 138(3), 581–613, 2018) an explicit analysis of best interpolation results in VE spaces, as well as stabilization estimates that are explicit in terms of the degree of accuracy p of the method. We prove exponential convergence of the hp-VEM for Stokes problems with regular right-hand sides. We corroborate the theoretical estimates with numerical tests for both the p- and hp-versions of the method.
  • Analytic regularity for the Navier-Stokes equations in polygons with mixed boundary conditions
    Item type: Report
    He, Yanchen; Marcati, Carlo; Schwab, Christoph (2021)
    SAM Research Report
    We prove weighted analytic regularity of Leray-Hopf variational solutions for the stationary, incompressible Navier-Stokes Equations (NSE) in plane polygonal domains, subject to analytic body forces. We admit mixed boundary conditions which may change type at each vertex, under the assumption that homogeneous Dirichlet (''no-slip'') boundary conditions are prescribed on at least one side at each vertex of the domain. The weighted analytic regularity results are established in Hilbertian Kondrat'ev spaces with homogeneous corner weights. The proofs rely on a priori estimates for the corresponding linearized boundary value problem in sectors in corner-weighted Sobolev spaces and on an induction argument for the weighted norm estimates on the quadratic nonlinear term in the NSE, in a polar frame.
Publications1 - 10 of 29