Convergence and superconvergence analyses of HDG methods for time fractional diffusion problems

Abstract

We study the hybridizable discontinuous Galerkin (HDG) method for the spatial discretization of time fractional diffusion models with Caputo derivative of order 0 < α < 1. For each time t ∈ [0, T ], the HDG approximations are taken to be piecewise polynomials of degree k ≥ 0 on the spatial domain Ω, the approximations to the exact solution u in the L ∞ 0, T ; L 2 (Ω)norm and to ∇u in the L ∞ 0, T ; L 2 (Ω) -norm are proven to converge with the rate h k+1 provided that u is sufficiently regular, where h is the maximum diameter of the elements of the mesh. Moreover, for k ≥ 1, we obtain a superconvergence result which allows us to compute, in an elementwise manner, a new approximation for u converging with a rate h k+2 (ignoring the logarithmic factor), for quasi-uniform spatial meshes. Numerical experiments validating the theoretical results are displayed.

The accuracy of an HDG method for conservative fractional diffusion equations

“…Hence, we need to show that the only solution to is λ h = 0. Taking μ = λ h in gives On the other hand, taking η = μ = λ h in implies Setting μ = λ h and in and summing over K ∈ Ω h , we get Inserting into and using the resulting equation in , we get that As in the proof of Lemma , the coercivity results in Mustapha and Schötzau, lemma 3.1(ii) and Mustapha et al, section 3, equation (9) imply the existence of a constant C > 0 such that $$$$…”

“…In Cockburn and Mustapha, the HDG method for numerically solving subdiffusion fractional model problem of order ‐ α with −1 < α < 0 is analyzed. The spatial discretization of time fractional diffusion equation with Caputo derivative of order 0 < α < 1 is studied by HDG method . The author, Celiker, and Kurulay introduced an HDG method for a Bagley‐Torvik equation with a Dirichlet boundary conditions and for a class of fractional boundary value problem in Karaaslan et al and Karaaslan et al, respectively.…”

Solvability of the fractional Volterra‐Fredholm integro differential equation by hybridizable discontinuous Galerkin method

“…The existence of the coercivity results in Mustapha and Schötzau and Mustapha et al 7,8 and the condition of the kernels shown in Theorem 3.1 are given such that…”

“…These methods are generally applied for ordinary or partial differential equation in the literature. The number of papers in which the method is used for fractional differential equations are limited [6][7][8][9][10][11] So far, the application of the HDG method for fractional integro differential equation has not been observed in the literature. This paper is divided into five sections.…”

The method of approximate particular solutions to simulate an anomalous mobile‐immobile transport process

“…So, numerical techniques have received an increasing attention for solution approximation of fractional order differential equations. Especially during the recent years, various numerical and computational methods, such as finite difference method, 14,15,[18][19][20] finite element method, 21,22 pseudo-spectral method, [23][24][25][26] and semi-analytical methods [27][28][29][30][31][32] have been developed and formulated for solving several types of constant and variable order fractional transport models. In recent decades, the meshless numerical methods have been introduced, developed, and well-used for solving many practical problems in applied sciences and engineering.…”