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Abstract We derive a residual-based a posteriori error estimator for the conforming hp -Adaptive Finite Element Method ( hp -AFEM) for the steady state Stokes problem describing the slow motion of an incompressible fluid. This error estimator is obtained by extending the idea of a posteriori error estimation for the classical h -version of AFEM. We also establish the reliability and efficiency of the error estimator. The proofs are based on the well-known Clément-type interpolation operator introduced in [28] in the context of the hp -AFEM. Numerical experiments show the performance of an adaptive hp-FEM algorithm using the proposed a posteriori error estimator.
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Algorithms for Parallel Generic hp -Adaptive Finite Element Software
Thehp-adaptive finite element method—where one independently chooses the mesh size (h) and polynomial degree (p) to be used on each cell—has long been known to have better theoretical convergence properties than eitherh- orp-adaptive methods alone. However, it is not widely used, owing at least in part to the difficulty of the underlying algorithms and the lack of widely usable implementations. This is particularly true when used with continuous finite elements. Herein, we discuss algorithms that are necessary for a comprehensive and generic implementation ofhp-adaptive finite element methods on distributed-memory, parallel machines. In particular, we will present a multistage algorithm for the unique enumeration of degrees of freedom suitable for continuous finite element spaces, describe considerations for weighted load balancing, and discuss the transfer of variable size data between processes. We illustrate the performance of our algorithms with numerical examples and demonstrate that they scale reasonably up to at least 16,384 message passage interface processes. We provide a reference implementation of our algorithms as part of the open source librarydeal.II.
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- PAR ID:
- 10490639
- Editor(s):
- na
- Publisher / Repository:
- ACM
- Date Published:
- Journal Name:
- ACM Transactions on Mathematical Software
- Volume:
- 49
- Issue:
- 3
- ISSN:
- 0098-3500
- Page Range / eLocation ID:
- 1 to 26
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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- Free Publicly Accessible Full Text
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Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.1145/3603372
