A painless multi-level automatic goal-oriented hp-adaptive coarsening strategy for elliptic and non-elliptic problems
This work extends an automatic energy-norm $hp$-adaptive strategy based on performing quasi-optimal unrefinements to the case of non-elliptic problems and goal-oriented adaptivity. The proposed approach employs a multi-level hierarchical data structure and alternates global $h$- and $p$-refinements...
| Autores: | , , , , |
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| Formato: | artículo |
| Estado: | Versión publicada |
| Fecha de publicación: | 2022 |
| País: | España |
| Recursos: | Basque Center for Applied Mathematics (BCAM) |
| Repositorio: | BIRD. BCAM's Institutional Repository Data |
| OAI Identifier: | oai:bird.bcamath.org:20.500.11824/1550 |
| Acesso em linha: | http://hdl.handle.net/20.500.11824/1550 |
| Access Level: | acceso abierto |
| Palavra-chave: | Goal-oriented adaptivity $hp$-adaptivity Finite element method Unrefinements non-Elliptic problems |
| Resumo: | This work extends an automatic energy-norm $hp$-adaptive strategy based on performing quasi-optimal unrefinements to the case of non-elliptic problems and goal-oriented adaptivity. The proposed approach employs a multi-level hierarchical data structure and alternates global $h$- and $p$-refinements with a coarsening step. Thus, at each unrefinement step, we eliminate the basis functions with the lowest contributions to the solution. When solving elliptic problems using energy-norm adaptivity, the removed basis functions are those with the lowest contributions to the energy of the solution. For non-elliptic problems or goal-oriented adaptivity, we propose an upper bound of the error representation expressed in terms of an inner product of the specific equation, leading to error indicators that deliver quasi-optimal $hp$-unrefinements. This unrefinement strategy removes unneeded unknowns possibly introduced during the pre-asymptotical regime. In addition, the grids over which we perform the unrefinements are arbitrary, and thus, we can limit their size and associated computational costs. We numerically analyze our algorithm for energy-norm and goal-oriented adaptivity. In particular, we solve two-dimensional ($2$D) Poisson, Helmholtz, convection-dominated equations, and a three-dimensional ($3$D) Helmholtz-like problem. In all cases, we observe \revb{exponential} convergence rates. Our algorithm is robust and straightforward to implement; therefore, it can be easily adapted for industrial applications. |
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