A Q1finite volume element scheme for anisotropic diffusion problems on general convex quadrilateral mesh Abstract   In this paper, we study a      Q    1     finite volume element scheme for anisotropic diffusion problems on general convex quadrilateral mesh. It is known that the coercivity is the basement for some other theoretical results (stability,      H    1      and      L    2      error estimates, etc.) and the existing results were mainly obtained on      h    1    xcexb3     parallelogram meshes with scalar diffusion coefficients. For the cases of full diffusion tensors and arbitrary convex quadrilateral meshes, we obtain a necessary and sufficient condition for the positive definiteness of the cell matrix related to the cell bilinear form. Based on this result, a sufficient condition is suggested to guarantee the coercivity of the scheme. More interesting is that, this sufficient condition covers the traditional      h    1    xcexb3     parallelogram mesh assumption and has an explicit expression, by which one can easily judge on any diffusion tensor and any mesh with arbitrary mesh size     h  u003e  0    . Moreover, an      H    1      error estimate is obtained without the      h    1    xcexb3     parallelogram assumption, and some numerical results are also provided to validate the theoretical results.

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.