Local convergence of the FEM for the integral fractional Laplacian

We provide for first order discretizations of the integral fractional Laplacian sharp local error estimates on proper subdomains in both the local $H^1$-norm and the localized energy norm. Our estimates have the form of a local best approximation error plus a global error measured in a weaker norm

Progress in Experimental Research of Turbine Aeroacoustics

Modifications on the intermediate turbine duct in order to reduce noise emissions by changing interaction frequencies and/or modes capable to propagate are presented. Also different turbine exit casings are described that are optimised to reduce interaction noise that is propagating through the engine and is one of the major noise sources during landing (operating point approach). The most promising modifications to reduce sound power levels are described. Depending on different modifications at specific operating points, the reduction of sound power level is between 5 dB and 10 dB, which is a significant reduction. However, some of these measures show an increase in aerodynamic losses. Therefore, a compromise has to be found between higher losses during a short duration (e.g. landing) and significant noise reduction. The chapter focuses on experimental results obtained in the test facilities of the Institute for Thermal Turbomachinery and Machine Dynamics at Graz University of Technology

Two-Level Error Estimation for the Integral Fractional Laplacian

For the singular integral definition of the fractional Laplacian, we consider an adaptive finite element method steered by two-level error indicators. For this algorithm, we show linear convergence in two and three space dimensions as well as convergence of the algorithm with optimal algebraic rates in 2D, when newest vertex bisection is employed for mesh refinement. A key step hereby is an equivalence of the nodal and Scott–Zhang interpolation operators in certain weighted L2 -norms

Weighted analytic regularity for the integral fractional Laplacian in polyhedra

We prove weighted analytic regularity of solutions to the Dirichlet problem for the integral fractional Laplacian in polytopal three-dimensional domains and 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.Comment: arXiv admin note: text overlap with arXiv:2112.0815