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Analytic regularity for the Navier-Stokes equations in polygons with mixed boundary conditions
(2021)SAM Research ReportWe 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 ...Report -
Exponential Convergence of Deep Operator Networks for Elliptic Partial Differential Equations
(2021)SAM Research ReportReport -
Weighted analytic regularity for the integral fractional Laplacian in polygons
(2021)SAM Research ReportWe prove weighted analytic regularity of solutions to the Dirichlet problem for the integral fractional Laplacian in polygons with analytic right-hand side. We localize the problem through the Caffarelli-Silvestre extension and study the tangential differentiability of the extended solutions, followed by bootstrapping based on Caccioppoli inequalities on dyadic decompositions of vertex, edge, and edge-vertex neighborhoods.Report -
Exponential convergence of hp-FEM for the integral fractional Laplacian in 1D
(2022)SAM Research ReportWe prove weighted analytic regularity for the solution of the integral frac tional 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.Report -
Exponential convergence of mixed hp-DGFEM for the incompressible Navier-Stokes equations in R²
(2020)SAM Research ReportIn 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 ...Report -
Exponential ReLU Neural Network Approximation Rates for Point and Edge Singularities
(2020)SAM Research ReportWe 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 ...Report -
Exponential Convergence of hp FEM for the Integral Fractional Laplacian in Polygons
(2022)SAM Research ReportReport -
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Expression Rates of Neural Operators for Linear Elliptic PDEs in Polytopes
(2024)SAM Research ReportWe study the approximation rates of a class of deep neural network approximations of operators, which arise as data-to-solution maps G † of linear elliptic partial differential equations (PDEs), and act between pairs X, Y of suitable infinite-dimensional spaces. We prove expression rate bounds for approximate neural operators G with the structure G = R ◦ A ◦ E, with linear encoders E and decoders R. The constructive proofs are via a ...Report