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In this paper, two finite difference/element approaches for the time-fractional subdiffusion equation with Dirichlet boundary conditions are developed, in which the time direction is approximated by the fractional linear multistep method and the space direction is approximated by the finite element method. The two methods are unconditionally stable and convergent of order O(τq +hr+1) in the L² norm, where q = 2-β or 2 when the analytical solution to the subdiffusion equation is sufficiently smooth, β (0 < β < 1) is the order of the fractional derivative, τ and h are the step sizes in time and space, respectively, and r is the degree of the polynomial space. The corresponding schemes for the subdiffusion equation with Neumann boundary conditions are presented as well, where the stability and convergence are shown. Numerical examples are provided to verify the theoretical analysis. Comparisons between the algorithms derived in this paper and the existing algorithms are given, which show that our numerical schemes exhibit better performances than the existing ones. [ABSTRACT FROM AUTHOR]

We consider in this paper the wave scattering problem by an object with Neumann boundary conditions in an anisotropic elastic body. To obtain an efficient numerical method (permitting the use of regular grids) we follow a fictitious domain approach coupled with a first order velocity stress formulation for elastodynamics. We first observe that the method does not always converge when the Qdiv1 - Q0 finite element is used. In particular, the method converges for some scattering object geometries but not for others. Note that the convergence of the Qdiv1 -Q0 finite element method was shown in [E. Bécache, P. Joly, and C. Tsogka, SIAM J. Numer. Anal., 39 (2002), pp. 2109-2132] for the elastodynamic problem in the absence of a scattering object (i.e., without the coupling of the mixed finite elements with the fictitious domain method). Therefore we propose here a modification of the Qdiv1 - Q0 element following the approach in [E. Bécache, J. Rodríguez, and C. Tsogka, On the convergence of the fictitious domain method for wave equation problems, Technical report 5802, INRIA, 2006], where the simpler acoustic case was considered. To study the numerical properties of the new element we carry out a dispersion analysis. Several numerical simulations as well as a numerical convergence analysis show that the proposed method provides a good approximate solution. [ABSTRACT FROM AUTHOR]