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Implementation of high-order, discontinuous Galerkin time stepping for fractional diffusion problems. (English) Zbl 1457.65129

In this paper, a practical implementation of the discontinuous Galerkin method is presented for the time integration of fractional diffusion problems with the accurate evaluation of certain coefficients. The expressions for the coefficients are chosen by Legendre polynomials as the shape functions employed in the DG time stepping. It was demonstrated that the high accuracy of the DG solution can be further improved by post-processing to form the reconstruction of the solution. For a classical diffusion problem, both the solution and the post-processed solution are quasi-optimal. It was shown in numerical experiments that the superconvergence properties of DG time-stepping for classical diffusion problems carry over to the fractional-order setting. Moreover, the jumps \([[U]]^{n-1}\) in the solution provide an easily computed error estimator for automatic step-size control.

MSC:

65M60 Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs
65D30 Numerical integration
65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs
42C10 Fourier series in special orthogonal functions (Legendre polynomials, Walsh functions, etc.)
35R11 Fractional partial differential equations
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References:

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