Convergence and superconvergence analyses of HDG methods for time fractional diffusion problems (Q273668)

This paper studies a numerical method resulting from using exact integration in time and a hybridizable discontinuous Galerkin (HDG) method for the spatial discretization of the time fractional diffusion model problem: \(^cD^{1-\alpha}u-\Delta u = f\) in \(\Omega\times (0,T]\), subject to \(u=g(x)\) in \(\partial\Omega\times (0,T]\) and \(u(x,0)=u_0(x)\) in \(\Omega\). Here, \(\Omega\subset\mathbb R^d\) \((1\leq d\leq 3)\) is a convex polyhedral domain, \(f\), \(g\) and \(u_0\) are smooth enough to ensure the existence of a solution \(u\in W^{1,1}(0,T; H^2(\Omega))\). When the HDG approximations are piecewise polynomials of degree \(k\geq 0\), it is shown that the approximations to the exact solution \(u\) and to its gradient \(\nabla u\) in the \(L^\infty\) norm converge with the rate \(h^{k+1}\), where \(h\) is the maximum diameter of the elements of the mesh. This result is further improved by the authors in case \(k\geq 1\). Numerical experiments are also presented in order to support the theoretical findings.