Communications on Applied Mathematics and Computation ›› 2019, Vol. 1 ›› Issue (3): 309-331.doi: 10.1007/s42967-019-00026-9

• ORIGINAL PAPER •    下一篇

An Adaptive hp-DG-FE Method for Elliptic Problems: Convergence and Optimality in the 1D Case

Paola Antonietti1, Claudio Canuto2, Marco Verani1   

  1. 1 MOX-Dipartimento di Matematica, Politecnico di Milano, P. zza Leonardo Da Vinci 32, 20133 Milan, Italy;
    2 Dipartimento di Scienze Matematiche, Politecnico di Torino, Corso Duca degli Abruzzi 24, 10129 Turin, Italy
  • 收稿日期:2018-05-31 修回日期:2018-12-30 出版日期:2019-09-20 发布日期:2019-09-09
  • 通讯作者: Claudio Canuto, Paola Antonietti, Marco Verani E-mail:claudio.canuto@polito.it;paola.antonietti@polimi.it;marco.verani@polimi.it
  • 基金资助:
    Work carried out within the "Progetto di Eccellenza 2018-2022", granted by MIUR (Italian Ministry of University and Research) to the Department of Mathematical Sciences, Politecnico di Torino. The authors are members of the INdAM research group GNCS, which granted partial support to this research.

An Adaptive hp-DG-FE Method for Elliptic Problems: Convergence and Optimality in the 1D Case

Paola Antonietti1, Claudio Canuto2, Marco Verani1   

  1. 1 MOX-Dipartimento di Matematica, Politecnico di Milano, P. zza Leonardo Da Vinci 32, 20133 Milan, Italy;
    2 Dipartimento di Scienze Matematiche, Politecnico di Torino, Corso Duca degli Abruzzi 24, 10129 Turin, Italy
  • Received:2018-05-31 Revised:2018-12-30 Online:2019-09-20 Published:2019-09-09
  • Supported by:
    Work carried out within the "Progetto di Eccellenza 2018-2022", granted by MIUR (Italian Ministry of University and Research) to the Department of Mathematical Sciences, Politecnico di Torino. The authors are members of the INdAM research group GNCS, which granted partial support to this research.

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摘要： We propose and analyze an hp-adaptive DG-FEM algorithm, termed hp-ADFEM, and its one-dimensional realization, which is convergent, instance optimal, and h- and p-robust. The procedure consists of iterating two routines:one hinges on Binev's algorithm for the adaptive hp-approximation of a given function, and finds a near-best hp-approximation of the current discrete solution and data to a desired accuracy; the other one improves the discrete solution to a finer but comparable accuracy, by iteratively applying Dörfler marking and h refinement.

关键词: Elliptic problem, Discontinuous Galerkin method, hp-adaptivity, Convergence and optimality

Abstract: We propose and analyze an hp-adaptive DG-FEM algorithm, termed hp-ADFEM, and its one-dimensional realization, which is convergent, instance optimal, and h- and p-robust. The procedure consists of iterating two routines:one hinges on Binev's algorithm for the adaptive hp-approximation of a given function, and finds a near-best hp-approximation of the current discrete solution and data to a desired accuracy; the other one improves the discrete solution to a finer but comparable accuracy, by iteratively applying Dörfler marking and h refinement.

Key words: Elliptic problem, Discontinuous Galerkin method, hp-adaptivity, Convergence and optimality

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