Fast summation by interval clustering for an evolution equation with memory

We solve a fractional diffusion equation using a piecewise-constant, discontinuous Galerkin method in time combined with a continuous, piecewise-linear finite element method in space. If there are $N$ time levels and $M$ spatial degrees of freedom, then a direct implementation of this method requires $O(N^2M)$ operations and $O(NM)$ active memory locations, owing to the presence of a memory term: at each time step, the discrete evolution equation involves a sum over \emph{all} previous time levels. We show how the computational cost can be reduced to $O(MN\log N)$ operations and $O(M\log N)$ active memory locations.Comment: 28 pages, 1 figur

Time-stepping error bounds for fractional diffusion problems with non-smooth initial data

We apply the piecewise constant, discontinuous Galerkin method to discretize a fractional diffusion equation with respect to time. Using Laplace transform techniques, we show that the method is first order accurate at the \$n\$th time level \$t_n\$, but the error bound includes a factor \$t_n^{-1}\$ if we assume no smoothness of the initial data. We also show that for smoother initial data the growth in the error bound as \$t_n\$ decreases is milder, and in some cases absent altogether. Our error bounds generalize known results for the classical heat equation and are illustrated for a model problem.Comment: 22 pages, 5 figure

Superconvergence of a discontinuous Galerkin method for fractional diffusion and wave equations

We consider an initial-boundary value problem for $\partial_tu-\partial_t^{-\alpha}\nabla^2u=f(t)$, that is, for a fractional diffusion ($-1<\alpha<0$) or wave ($0<\alpha<1$) equation. A numerical solution is found by applying a piecewise-linear, discontinuous Galerkin method in time combined with a piecewise-linear, conforming finite element method in space. The time mesh is graded appropriately near $t=0$, but the spatial mesh is quasiuniform. Previously, we proved that the error, measured in the spatial $L_2$-norm, is of order $k^{2+\alpha_-}+h^2\ell(k)$, uniformly in $t$, where $k$ is the maximum time step, $h$ is the maximum diameter of the spatial finite elements, $\alpha_-=\min(\alpha,0)\le0$ and $\ell(k)=\max(1,|\log k|)$. Here, we generalize a known result for the classical heat equation (i.e., the case $\alpha=0$) by showing that at each time level $t_n$ the solution is superconvergent with respect to $k$: the error is of order $(k^{3+2\alpha_-}+h^2)\ell(k)$. Moreover, a simple postprocessing step employing Lagrange interpolation yields a superconvergent approximation for any $t$. Numerical experiments indicate that our theoretical error bound is pessimistic if $\alpha<0$. Ignoring logarithmic factors, we observe that the error in the DG solution at $t=t_n$, and after postprocessing at all $t$, is of order $k^{3+\alpha_-}+h^2$.Comment: 24 pages, 2 figure

The use of Bioceramics as root-end filling materials in periradicular surgery: a literature review

Introduction: Periradicular surgery involves the placement of a root-end filling following root-end resection, to provide an apical seal to the root canal system. Historically several materials have been used in order to achieve this seal. Recently a class of materials known as Bioceramics have been adopted. The aim of this article is to provide a review of the outcomes of periradicular surgery when Bioceramic root-end filling materials are used on human permanent teeth in comparison to “traditional” materials. Methods &amp; results: An electronic literature search was performed in the databases of Web of Science, PubMed and Google Scholar, between 2006 and 2017, to collect clinical studies where Bioceramic materials were utilised as retrograde filling materials, and to compare such materials with traditional materials. In this search, 1 systematic review and 14 clinical studies were identified. Of these, 8 reported the success rates of retrograde Bioceramics, and 6 compared treatment outcomes of mineral trioxide aggregate (MTA) and traditional cements when used as root-end filling materials. Conclusion: Bioceramic root-end filling materials are shown to have success rates of 86.4–95.6% (over 1–5 years). Bioceramics has significantly higher success rates than amalgam, but they were statistically similar to intermediate restorative material (IRM) and Super ethoxybenzoic acid (Super EBA) when used as retrograde filling materials in apical surgery. However, it seems that the high success rates were not solely attributable to the type of the root-end filling materials. The surgical/microsurgical techniques and tooth prognostic factors may significantly affect treatment outcome