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Two-grid hp-version discontinuous Galerkin finite element methods for quasilinear elliptic PDEs on agglomerated coarse meshes

Abstract

This article considers the extension of two-grid hp-version discontinuous Galerkin finite element methods for the numerical approximation of second-order quasilinear elliptic boundary value problems of monotone type to the case when agglomerated polygonal/polyhedral meshes are employed for the coarse mesh approximation. We recall that within the two-grid setting, while it is necessary to solve a nonlinear problem on the coarse approximation space, only a linear problem must be computed on the original fine finite element space. In this article, the coarse space will be constructed by agglomerating elements from the original fine mesh. Here, we extend the existing a priori and a posteriori error analysis for the two-grid hp-version discontinuous Galerkin finite element method from Congreve et al. [1] for coarse meshes consisting of standard element shapes to include arbitrarily agglomerated coarse grids. Moreover, we develop an hp-adaptive two-grid algorithm to adaptively design the fine and coarse finite element spaces; we stress that this is undertaken in a fully automatic manner, and hence can be viewed as blackbox solver. Numerical experiments are presented for two- and three-dimensional problems to demonstrate the computational performance of the proposed hp-adaptive two-grid method.

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Funding

SC has been supported by Charles University Research program no. UNCE/SCI/023 and the Czech Science Foundation (GAČR) project no. 20-01074S. PH received financial support from the EPSRC under the grant EP/R030707/1.

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Correspondence to Scott Congreve.

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PH is a member of the editorial board.

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Communicated by Long Chen

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Congreve, S., Houston, P. Two-grid hp-version discontinuous Galerkin finite element methods for quasilinear elliptic PDEs on agglomerated coarse meshes. Adv Comput Math 48, 54 (2022). https://doi.org/10.1007/s10444-022-09968-w

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  • DOI: https://doi.org/10.1007/s10444-022-09968-w

Keywords

  • Discontinuous Galerkin finite element methods
  • Polytopic elements
  • hp-finite element methods
  • Two-grid methods
  • Quasilinear PDEs

Mathematics Subject Classification (2010)

  • 35J62
  • 65N30
  • 65N50