On the Convergence of the Local Discontinuous Galerkin Method Applied to a Stationary One Dimensional Fractional Diffusi
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On the Convergence of the Local Discontinuous Galerkin Method Applied to a Stationary One Dimensional Fractional Diffusion Problem P. Castillo1
· S. Gómez2
Received: 18 December 2019 / Revised: 8 August 2020 / Accepted: 8 October 2020 © Springer Science+Business Media, LLC, part of Springer Nature 2020
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 < α < 2. Well posedness of the stabilized and non stabilized LDG method is proved. Using a penalty term of order O h 1−α 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 ofconvergence for all fractional orders. A numerical study suggests a less restrictive, O h −α , spectral condition number of the stiffness matrix by using the proposed penalty term compared to the O h −2 −1 penalization term is chosen. The sharpness growth obtained when the traditional O h 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. Keywords Local discontinuous Galerkin method · Riesz and Riemann–Liouville operators · Fractional diffusion Mathematics Subject Classification 65M60 · 65M08 · 65-05
1 Introduction Fractional order derivatives originated as a mathematical curiosity almost three centuries ago, however supported by a vast physical evidence, their use has become a standard tool in
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S. Gómez [email protected] P. Castillo [email protected]
1
Department of Mathematical Sciences, University of Puerto Rico, Call Box 9000, Mayagüez 00681, Puerto Rico
2
Department of Mathematics, University of Pavia, Via Ferrata 5, 27100 Pavia, Italy 0123456789().: V,-vol
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Page 2 of 22
Journal of Scientific Computing
(2020) 85:32
mathematical modeling. A collection of diverse fractional calculus applications in science and engineering can be found in the recent article of Sun et al. [34] and references therein. In this article we are interested in the numerical approximation of the solution of the general two point boundary value problem − ∂ α u/∂|x|α = f , in = (a, b),
(1.1a)
u = 0, in R \ ,
(1.1b)
where ∂ α (·)/∂|x|α is the fractional Riesz operator of order 1 < α < 2. Contrary to the classical Poisson’s equation (α = 2), close form solutions for the model problem (1.1) are scarce, even for simple source terms. Over the last decade there has been an increasing interest in the numerical solution of partial differential equations with fractional derivatives. Due to its simple implementation, finite difference methods have been commonly used in the approximation of problems with fractional derivati
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