Some error estimates for the finite volume element method for a parabolic problem

We study spatially semidiscrete and fully discrete finite volume element methods for the homogeneous heat equation with homogeneous Dirichlet boundary conditions and derive error estimates for smooth and nonsmooth initial data. We show that the results of our earlier work \cite{clt11} for the lumped mass method carry over to the present situation. In particular, in order for error estimates for initial data only in $L_2$ to be of optimal second order for positive time, a special condition is required, which is satisfied for symmetric triangulations. Without any such condition, only first order convergence can be shown, which is illustrated by a counterexample. Improvements hold for triangulations that are almost symmetric and piecewise almost symmetric

Two fully discrete schemes for fractional diffusion and diffusion-wave equations with nonsmooth data

We consider initial/boundary value problems for the subdiffusion and diffusion-wave equations involving a Caputo fractional derivative in time. We develop two fully discrete schemes based on the piecewise linear Galerkin finite element method in space and convolution quadrature in time with the generating function given by the backward Euler method/second-order backward difference method, and establish error estimates optimal with respect to the regularity of problem data. These two schemes are first- and second-order accurate in time for both smooth and nonsmooth data. Extensive numerical experiments for two-dimensional problems confirm the convergence analysis and robustness of the schemes with respect to data regularity. Read More: http://epubs.siam.org/doi/10.1137/14097956

Variational formulation of problems involving fractional order differential operators

In this work, we consider boundary value problems involving either Caputo or Riemann-Liouville fractional derivatives of order α ∈ (1, 2) on the unit interval (0, 1). These fractional derivatives lead to nonsymmetric boundary value problems, which are investigated from a variational point of view. The variational problem for the Riemann-Liouville case is coercive on the space Hα/2 0 (0, 1) but the solutions are less regular, whereas that for the Caputo case involves different test and trial spaces. The numerical analysis of these problems requires the so-called shift theorems which show that the solutions of the variational problem are more regular. The regularity pickup enables one to establish convergence rates of the finite element approximations. The analytical theory is then applied to the Sturm-Liouville problem involving a fractional derivative in the leading term. Finally, extensive numerical results are presented to illustrate the error estimates for the source problem and eigenvalue problem

A Posteriori Error Estimates for Nonconforming Approximations of Evolutionary Convection-Diffusion Problems

We derive computable upper bounds for the difference between an exact solution of the evolutionary convection-diffusion problem and an approximation of this solution. The estimates are obtained by certain transformations of the integral identity that defines the generalized solution. These estimates depend on neither special properties of the exact solution nor its approximation, and involve only global constants coming from embedding inequalities. The estimates are first derived for functions in the corresponding energy space, and then possible extensions to classes of piecewise continuous approximations are discussed.Comment: 10 page

The Role of Water in the Stability of Cratonic Keels

Cratons are typically underlain by large, deep, and old lithospheric keels (to greater than 200 km depth, greater than 2.5 Ga old) projecting into the asthenosphere (e.g., Jordan, 1978; Richardson et al., 1984). This has mystified Earth scientists as the dynamic and relatively hot asthenosphere should have eroded away these keels over time (e.g., Sleep, 2003; O'Neill et al., 2008; Karato, 2010). Three key factors have been invoked to explain cratonic root survival: 1) Low density makes the cratonic mantle buoyant (e.g., Poudjom Djomani et al., 2001). 2) Low temperatures (e.g., Pollack, 1986; Boyd, 1987), and 3) low water contents (e.g., Pollack, 1986), would make cratonic roots mechanically strong. Here we address the mechanism of the longevity of continental mantle lithosphere by focusing on the water parameter. Although nominally anhydrous , olivine, pyroxene and garnet can accommodate trace amounts of water in the form of H bonded to structural O in mineral defects (e.g., Bell and Rossman, 1992). Olivine softens by orders of magnitude if water (1-1000 ppm H2O) is added to its structure (e.g., Mackwell et al., 1985). Our recent work has placed constraints on the distribution of water measured in peridotite minerals in the cratonic root beneath the Kaapvaal in southern Africa (Peslier et al., 2010). At P greater than 5 GPa, the water contents of pyroxene remain relatively constant while those of olivine systematically decrease from 50 to less than 10 ppm H2O at 6.4 GPa. We hypothesized that at P greater than 6.4 GPa, i.e. at the bottom of the cratonic lithosphere, olivines are essentially dry (greater than 10 ppm H2O). As olivine likely controls the rheology of the mantle, we calculated that the dry olivines could be responsible for a contrast in viscosity between cratonic lithosphere and surrounding asthenosphere large enough to explain the resistance of cratonic root to asthenospheric delamination