hp-FEM for the integral fractional Laplacian: quadrature
A talk in the Nonlocal Equations: Analysis and Numerics series by
Markus Melenk from Wien
| Abstract: | For the Dirichlet problem of the integral fractional Laplacian on intervals
$\Omega$ and on polygons $\Omega$, it has recently been shown that exponential convergence of the hp-FEM based on suitably designed meshes can be achieved,
[Faustmann, Marcati, Melenk, Schwab, 2023]. These meshes are geometrically refined towards the edges and corners of $\Omega$. The geometric refinement
towards the edges results in anisotropic meshes away from corners. The
use of such anisotropic elements is crucial for the exponential convergence
result.
In this talk, we address the issue of setting the stiffness matrix. We
show that a judicious combination of Duffy-like transformations and hp-quadrature techniques allow one to set up the matrix with work growing
algebraically in the problem size while retaining the
exponential convergence of hp-FEM. The emphasis will be placed on the
1D fractional Laplacian, [Bahr, Faustmann, Melenk, 2024]. Within the CRC this talk is associated to the project(s): A7 |