A hybridizable discontinuous Galerkin method for fractional diffusion problems

Abstract

We study the use of the hybridizable discontinuous Galerkin (HDG) method for numerically solving fractional diffusion equations of order −α with −1 < α < 0. For exact time-marching, we derive optimal algebraic error estimates assuming that the exact solution is sufficiently regular. Thus, if for each time t ∈ [0, T ] the approximations are taken to be piecewise polynomials of degree k ≥ 0 on the spatial domain Ω , the approximations to 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 , where h is the maximum diameter of the elements of the mesh. Moreover, for k ≥ 1 and quasi-uniform meshes, we obtain a superconvergence result which allows us to compute, in an elementwise manner, a new approximation for u converging with a rate of log(T h −2/(α+1) ) h k+2 .

Numerical solution of time fractional Burgers equation

“…Applications of differentiation and integration with non-integer orders can be traced back to premature in history, so it can be said that it is not new [16]. Many different techniques and methods of dealing with fractional differential equations resulting analytical and numerical solutions can be found in a wide variety of studies in the literature [17,18,19,20,21,22,23,24,25,26,27,28,29,30,31].…”

Solving 2D time‐fractional diffusion equations by a pseudospectral method and Mittag‐Leffler function evaluation

“…where`i j .x/ is the Lagrangian basis function based on the Chebyshev points (15). Because the solution u.x ij , t/ for i D 0, n and j D 0, n is already known from the homogeneous boundary condition (8), it is not necessary to consider these values of i and j.…”

New analytical tools for HDG in elasticity, with applications to elastodynamics

“…It uses only polynomial basis functions, achieves one order higher convergence rate for the displacement without post-processing, and its computational complexity is the same to the standard HDG method (order k for both stress and displacement) for their global systems. However, the existing error analysis of HDG+ methods are all based on using orthogonal projections [15,18,19,20,21], which make the analysis slightly more complicated (it requires a bootstrapping argument to prove convergence of all variables, as opposed to consecutive energy and duality proofs), and detached from the existing projection-based error analysis of HDG methods [3,6,7,8,9,10,11,14], where specifically constructed projections are used to make the analysis simple and concise. This motivates us to find a new kind of projection for HDG+ for elasticity.…”