POLYGONAL DISCONTINUOUS GALERKIN METHODS ON CURVED REGIONS AND ITS MULTIGRID PRECONDITIONER

Liu Yi, Wang Yanqiu

Mathematica Numerica Sinica ›› 2022, Vol. 44 ›› Issue (3) : 396-421.

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Mathematica Numerica Sinica ›› 2022, Vol. 44 ›› Issue (3) : 396-421. DOI: 10.12286/jssx.j2021-0792
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POLYGONAL DISCONTINUOUS GALERKIN METHODS ON CURVED REGIONS AND ITS MULTIGRID PRECONDITIONER

  • Liu Yi, Wang Yanqiu
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Abstract

The main purpose of this paper is to study the discontinuous Galerkin discretization of second-order elliptic partial differential equations on curved regions. We use multiple short edges to approximate the curved boundary and achieve the optimal convergence order in H1 norm for high-order elements. This method is also applied to the interface problem with curved interfaces and obtains the optimal convergence of high-order elements. Furthermore, we prove that the W-cycle and V-cycle multigrid preconditioners converge uniformly provided that the number of smoothing is sufficiently large. Finally, numerical results are presented to verify the correctness of the theoretical results.

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curved boundary / curved interface / polygonal mesh / interior penalty discontinuous Galerkin method / multigrid

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Liu Yi, Wang Yanqiu. POLYGONAL DISCONTINUOUS GALERKIN METHODS ON CURVED REGIONS AND ITS MULTIGRID PRECONDITIONER. Mathematica Numerica Sinica, 2022, 44(3): 396-421 https://doi.org/10.12286/jssx.j2021-0792

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