Discrete $p$-robust $\mathbf{H}(\mathrm{div})$-liftings and a posteriori estimates for elliptic problems with $H^{-1}$ source terms
Fuente:
arXiv
Salvato in:
| Autori principali: | , , |
|---|---|
| Natura: | Preprint |
| Pubblicazione: |
2016
|
| Soggetti: | |
| Accesso online: | |
| Tags: |
Aggiungi Tag
Nessun Tag, puoi essere il primo ad aggiungerne!!
|
| _version_ | 1866910801817763840 |
|---|---|
| author | Ern, Alexandre Smears, Iain Vohralík, Martin |
| author_facet | Ern, Alexandre Smears, Iain Vohralík, Martin |
| contents | We establish the existence of liftings into discrete subspaces of $\mathbf{H}(\mathrm{div})$ of piecewise polynomial data on locally refined simplicial partitions of polygonal/polyhedral domains. Our liftings are robust with respect to the polynomial degree. This result has important applications in the a posteriori error analysis of parabolic problems, where it permits the removal of so-called transition conditions that link two consecutive meshes. It can also be used in a the posteriori error analysis of elliptic problems, where it allows the treatment of meshes with arbitrary numbers of hanging nodes between elements. We present a constructive proof based on the a posteriori error analysis of an auxiliary elliptic problem with $H^{-1}$ source terms, thereby yielding results of independent interest. In particular, for such problems, we obtain guaranteed upper bounds on the error along with polynomial-degree robust local efficiency of the estimators. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1610_01786 |
| institution | arXiv |
| publishDate | 2016 |
| record_format | arxiv |
| spellingShingle | Discrete $p$-robust $\mathbf{H}(\mathrm{div})$-liftings and a posteriori estimates for elliptic problems with $H^{-1}$ source terms Ern, Alexandre Smears, Iain Vohralík, Martin Numerical Analysis 65N30 We establish the existence of liftings into discrete subspaces of $\mathbf{H}(\mathrm{div})$ of piecewise polynomial data on locally refined simplicial partitions of polygonal/polyhedral domains. Our liftings are robust with respect to the polynomial degree. This result has important applications in the a posteriori error analysis of parabolic problems, where it permits the removal of so-called transition conditions that link two consecutive meshes. It can also be used in a the posteriori error analysis of elliptic problems, where it allows the treatment of meshes with arbitrary numbers of hanging nodes between elements. We present a constructive proof based on the a posteriori error analysis of an auxiliary elliptic problem with $H^{-1}$ source terms, thereby yielding results of independent interest. In particular, for such problems, we obtain guaranteed upper bounds on the error along with polynomial-degree robust local efficiency of the estimators. |
| title | Discrete $p$-robust $\mathbf{H}(\mathrm{div})$-liftings and a posteriori estimates for elliptic problems with $H^{-1}$ source terms |
| topic | Numerical Analysis 65N30 |
| url | https://arxiv.org/abs/1610.01786 |