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On the Convergence of the Local Discontinuous Galerkin Method Applied to a Stationary One Dimensional Fractional Diffusion Problem

Abstract

The mixed formulation of the Local Discontinuous Galerkin (LDG) method is presented for a two boundary value problem that involves the Riesz operator with fractional order \(1< \alpha < 2\). Well posedness of the stabilized and non stabilized LDG method is proved. Using a penalty term of order \({{\mathcal {O}}}\left( h^{1-\alpha }\right) \) a sharp error estimate in a mesh dependent energy semi-norm is developed for sufficiently smooth solutions. Error estimates in the \(L^2\)-norm are obtained for two auxiliary variables which characterize the LDG formulation. Our analysis indicates that the non stabilized version of the method achieves higher order of convergence for all fractional orders. A numerical study suggests a less restrictive, \({{\mathcal {O}}}\left( h^{-\alpha }\right) \), spectral condition number of the stiffness matrix by using the proposed penalty term compared to the \({{\mathcal {O}}}\left( h^{-2}\right) \) growth obtained when the traditional \({{\mathcal {O}}}\left( h^{-1}\right) \) penalization term is chosen. The sharpness of our error estimates is numerically validated with a series of numerical experiments. The present work is the first attempt to elucidate the main differences between both versions of the method.

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Acknowledgements

We kindly thank the anonymous reviewers for their valuable suggestions to improve the content of this work and its final presentation.

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Correspondence to S. Gómez.

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Castillo, P., Gómez, S. On the Convergence of the Local Discontinuous Galerkin Method Applied to a Stationary One Dimensional Fractional Diffusion Problem. J Sci Comput 85, 32 (2020). https://doi.org/10.1007/s10915-020-01335-5

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Keywords

  • Local discontinuous Galerkin method
  • Riesz and Riemann–Liouville operators
  • Fractional diffusion

Mathematics Subject Classification

  • 65M60
  • 65M08
  • 65-05