FEM-BEM coupling in Fractional Diffusion
A talk in the Nonlocal Equations: Analysis and Numerics 2023 series by
Markus Faustmann from Wien
| Abstract: | In this talk, we consider fractional differential equations posed on the full space $\mathbb{R}^d$. Employing the well-known Caffarelli-Silvestre extension to $\mathbb{R}^d\times \mathbb{R}^+$, well-posedness of a weak formulation follows from Poincaré-type estimates. In order to derive a computable numerical approximation, we start with a truncation in the extended variable, i.e., we consider the extension problem on $\mathbb{R}^d\times (0,\mathcal{Y})$ for some $\mathcal{Y}>0$. Then, a natural question is, if and how fast solutions to the truncated problem converge to the full-space solution as $\mathcal{Y}\rightarrow \infty$. Using purely variational techniques, we derive an algebraic rate of decay as well as estimates for higher order derivatives of the truncated extension problem. These decay and regularity estimates can now be used to derive a-priori estimates for the error between the exact full-space solution and an approximation based on a coupling of finite elements and boundary elements. More precisely, this approximation is based on a diagonalization procedure from [1] that leads to a sequence of scalar Helmholtz-type problems, which are discretized with a symmetric FEM-BEM coupling. Combined with a $hp$-FEM discretization in the extended variable, this gives a full computable approximation. [1] L. Banjai, J. M. Melenk, R. H. Nochetto, E. Otárola, A. J. Salgado, and C. Schwab. Tensor FEM for spectral fractional diffusion. Found. Comput. Math., 19(4):901–962, 2019. [2] M.Faustmann, A.Rieder, Fractional Diffusion in the full space: decay and regularity, arXiv:2301.05503. [3] M.Faustmann, A.Rieder, FEM-BEM coupling in Fractional Diffusion, to appear soon. Within the CRC this talk is associated to the project(s): A7 |