A locally conservative staggered least squares method on polygonal meshes

Lina Zhao; Eun-Jae Park; Lina Zhao; Eun-Jae Park

doi:10.3934/mine.2024014

Mathematics in Engineering

2024, Volume 6, Issue 2: 339-362. doi: 10.3934/mine.2024014

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A locally conservative staggered least squares method on polygonal meshes

Lina Zhao ¹,
Eun-Jae Park ^{2
,
,}

1.
Department of Mathematics, City University of Hong Kong, Hong Kong, China
2.
Department of Computational Science and Engineering, Yonsei University, Seoul 03722, Korea

Received: 30 June 2023 Revised: 20 March 2024 Accepted: 21 March 2024 Published: 26 March 2024

In this paper, we propose a novel staggered least squares method for elliptic equations on polygonal meshes. Our new method can be flexibly applied to rough grids and allows hanging nodes, which is of particular interest in practical applications. Moreover, it offers the advantage of not having to deal with inf-sup conditions and yielding positive definite discrete problems. Optimal a priori error estimates in energy norm are derived. In addition, a superconvergent estimates in energy norm are also developed by employing variational error expansion. The main difficulty involved here is to show the $ L^2 $ norm error estimates for the potential variable, where duality argument and the superconvergent estimates are the key ingredients. The single valued flux over the outer boundary of the dual partition enables us to construct a locally conservative flux. Numerical experiments confirm the theoretical findings and the performance of the adaptive mesh refinement guided by the least squares functional estimator are also displayed.
- staggered grid,
- least squares,
- hanging nodes,
- error estimates,
- superconvergence,
- local conservation,
- adaptive mesh refinement,
- general meshes
Citation: Lina Zhao, Eun-Jae Park. A locally conservative staggered least squares method on polygonal meshes[J]. Mathematics in Engineering, 2024, 6(2): 339-362. doi: 10.3934/mine.2024014

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Abstract

In this paper, we propose a novel staggered least squares method for elliptic equations on polygonal meshes. Our new method can be flexibly applied to rough grids and allows hanging nodes, which is of particular interest in practical applications. Moreover, it offers the advantage of not having to deal with inf-sup conditions and yielding positive definite discrete problems. Optimal a priori error estimates in energy norm are derived. In addition, a superconvergent estimates in energy norm are also developed by employing variational error expansion. The main difficulty involved here is to show the $ L^2 $ norm error estimates for the potential variable, where duality argument and the superconvergent estimates are the key ingredients. The single valued flux over the outer boundary of the dual partition enables us to construct a locally conservative flux. Numerical experiments confirm the theoretical findings and the performance of the adaptive mesh refinement guided by the least squares functional estimator are also displayed.

References

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