Superconvergent Functional Estimates from TensorProduct Generalized SummationbyParts Discretizations in Curvilinear Coordinates We investigate superconvergent functional estimates in curvilinear coordinates for diagonalnorm tensorproduct generalized summationbyparts operators. We show that interpolationextrapolation operators of degree greater than or equal to 2p are required to preserve at least 2p quadrature accuracy and functional superconvergence in curvilinear coordinates when: (1) the Jacobian of the coordinate transformation is approximated by the same generalized summationbyparts operator that is used to approximate the flux terms and (2) the degree of the generalized summationbyparts operator is lower than the degree of the polynomial used to represent the geometry of interest. Legendrexe2x80x93Gaussxe2x80x93Lobatto and Legendrexe2x80x93Gauss elementtype operators are considered. When the aforementioned condition (2) is violated for the Legendrexe2x80x93Gauss operators, there is an evenxe2x80x93odd quadrature convergence pattern that is explained by the cancellation of the leading truncation error terms for the interpolationextrapolation operators that correspond to the odddegree Legendrexe2x80x93Gauss operators. The theory developed is confirmed through numerical examples with a steady onedimensional problem and the unsteady twodimensional linear convection equation.

Superconvergence of a Finite Element Method for the Multiterm TimeFractional Diffusion Problem A timefractional initialboundary value problem is considered, where the differential equation has a sum of fractional derivatives of different orders, and the spatial domain lies in (mathbb Rd) with (din 1,2,3). A priori bounds on the solution and its derivatives are stated; these show that typical solutions have a weak singularity at the initial time (t0). A standard finite element method with mapped piecewise bilinears is used to discretise the spatial derivatives, while for each time derivative we use the L1 scheme on a temporal graded mesh. Our analysis reveals the optimal grading that one should use for this mesh. A novel discrete fractional Gronwall inequality is proved: the statement of this inequality and its proof are different from any previously published Gronwall inequality. This inequality is used to derive an optimal error estimate in (Linfty (H1)). It is also used to show that, if each mesh element is rectangular in the case (d2) or cubical in the case (d3), with the sides of the element parallel to the coordinate axes, then a simple postprocessing of the computed solution will yield a higher order of convergence in the spatial direction. Numerical results are presented to show the sharpness of our theoretical results.