Exponential Convergence of $hp$ FEM for Spectral Fractional Diffusion in Polygons
Fuente:
arXiv
Salvato in:
| Autori principali: | , , |
|---|---|
| Natura: | Preprint |
| Pubblicazione: |
2020
|
| Soggetti: | |
| Accesso online: | |
| Tags: |
Aggiungi Tag
Nessun Tag, puoi essere il primo ad aggiungerne!!
|
| _version_ | 1866914883749019648 |
|---|---|
| author | Banjai, Lehel Melenk, Jens M. Schwab, Christoph |
| author_facet | Banjai, Lehel Melenk, Jens M. Schwab, Christoph |
| contents | For the spectral fractional diffusion operator of order $2s\in (0,2)$ in bounded, curvilinear polygonal domains $Ω$ we prove exponential convergence of two classes of $hp$ discretizations under the assumption of analytic data, without any boundary compatibility, in the natural fractional Sobolev norm $\mathbb{H}^s(Ω)$. The first $hp$ discretization is based on writing the solution as a co-normal derivative of a $2+1$-dimensional local, linear elliptic boundary value problem, to which an $hp$-FE discretization is applied. A diagonalization in the extended variable reduces the numerical approximation of the inverse of the spectral fractional diffusion operator to the numerical approximation of a system of local, decoupled, second order reaction-diffusion equations in $Ω$. Leveraging results on robust exponential convergence of $hp$-FEM for second order, linear reaction diffusion boundary value problems in $Ω$, exponential convergence rates for solutions $u\in \mathbb{H}^s(Ω)$ of $\mathcal{L}^s u = f$ follow. Key ingredient in this $hp$-FEM are boundary fitted meshes with geometric mesh refinement towards $\partialΩ$.
The second discretization is based on exponentially convergent sinc quadrature approximations of the Balakrishnan integral representation of $\mathcal{L}^{-s}$, combined with $hp$-FE discretizations of a decoupled system of local, linear, singularly perturbed reaction-diffusion equations in $Ω$. The present analysis for either approach extends to polygonal subsets $\widetilde{\mathcal{M}}$ of analytic, compact $2$-manifolds $\mathcal{M}$. Numerical experiments for model problems in nonconvex polygonal domains and with incompatible data confirm the theoretical results.
Exponentially small bounds on Kolmogoroff $n$-widths of solutions sets for spectral fractional diffusion in polygons are deduced. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2011_05701 |
| institution | arXiv |
| publishDate | 2020 |
| record_format | arxiv |
| spellingShingle | Exponential Convergence of $hp$ FEM for Spectral Fractional Diffusion in Polygons Banjai, Lehel Melenk, Jens M. Schwab, Christoph Numerical Analysis 26A33, 65N12, 65N30 For the spectral fractional diffusion operator of order $2s\in (0,2)$ in bounded, curvilinear polygonal domains $Ω$ we prove exponential convergence of two classes of $hp$ discretizations under the assumption of analytic data, without any boundary compatibility, in the natural fractional Sobolev norm $\mathbb{H}^s(Ω)$. The first $hp$ discretization is based on writing the solution as a co-normal derivative of a $2+1$-dimensional local, linear elliptic boundary value problem, to which an $hp$-FE discretization is applied. A diagonalization in the extended variable reduces the numerical approximation of the inverse of the spectral fractional diffusion operator to the numerical approximation of a system of local, decoupled, second order reaction-diffusion equations in $Ω$. Leveraging results on robust exponential convergence of $hp$-FEM for second order, linear reaction diffusion boundary value problems in $Ω$, exponential convergence rates for solutions $u\in \mathbb{H}^s(Ω)$ of $\mathcal{L}^s u = f$ follow. Key ingredient in this $hp$-FEM are boundary fitted meshes with geometric mesh refinement towards $\partialΩ$. The second discretization is based on exponentially convergent sinc quadrature approximations of the Balakrishnan integral representation of $\mathcal{L}^{-s}$, combined with $hp$-FE discretizations of a decoupled system of local, linear, singularly perturbed reaction-diffusion equations in $Ω$. The present analysis for either approach extends to polygonal subsets $\widetilde{\mathcal{M}}$ of analytic, compact $2$-manifolds $\mathcal{M}$. Numerical experiments for model problems in nonconvex polygonal domains and with incompatible data confirm the theoretical results. Exponentially small bounds on Kolmogoroff $n$-widths of solutions sets for spectral fractional diffusion in polygons are deduced. |
| title | Exponential Convergence of $hp$ FEM for Spectral Fractional Diffusion in Polygons |
| topic | Numerical Analysis 26A33, 65N12, 65N30 |
| url | https://arxiv.org/abs/2011.05701 |