arXiv subjects

Kassem Mustapha

Publications and source records attributed to Kassem Mustapha.

25 records · Page 2Linked to original sources

arXiv2014

Convergence and superconvergence analyses of HDG methods for time fractional diffusion problems

We study the hybridizable discontinuous Galerkin (HDG) method for the spatial discretization of time fractional diffusion models with Caputo derivative of order $0<α<1$. For each time $t \in [0,T]$, the HDG approximations are taken to be piecewise polynomials of degree $k\ge0$ on the spatial domain~$Ω$, the approximations to the exact solution $u$ in the $L_\infty\bigr(0,T;L_2(Ω)\bigr)$-norm and to $\nabla u$ in the $L_\infty\bigr(0,T;{\bf L}_2(Ω)\bigr)$-norm are proven to converge with the rate $h^{k+1}$ provided that $u$ is sufficiently regular, where $h$ is the maximum diameter of the elements of the mesh. Moreover, for $k\ge1$, we obtain a superconvergence result which allows us to compute, in an elementwise manner, a new approximation for $u$ converging with a rate $h^{k+2}$ (ignoring the logarithmic factor), for quasi-uniform spatial meshes. Numerical experiments validating the theoretical results are displayed.

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arXiv2014

A hybridizable discontinuous Galerkin method for fractional diffusion problems

We study the use of the hybridizable discontinuous Galerkin (HDG) method for numerically solving fractional diffusion equations of order $-α$ with $-1<α<0$. For exact time-marching, we derive optimal algebraic error estimates {assuming} that the exact solution is sufficiently regular. Thus, if for each time $t \in [0,T]$ the approximations are taken to be piecewise polynomials of degree $k\ge0$ on the spatial domain~$Ω$, the approximations to $u$ in the $L_\infty\bigr(0,T;L_2(Ω)\bigr)$-norm and to $\nabla u$ in the $L_\infty\bigr(0,T;{\bf L}_2(Ω)\bigr)$-norm are proven to converge with the rate $h^{k+1}$, where $h$ is the maximum diameter of the elements of the mesh. Moreover, for $k\ge1$ and quasi-uniform meshes, we obtain a superconvergence result which allows us to compute, in an elementwise manner, a new approximation for $u$ converging with a rate of $\sqrt{\log(T h^{-2/(α+1)})}\, \,h^{k+2}$.

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arXiv2014

Time-stepping discontinuous Galerkin methods for fractional diffusion problems

Time-stepping $hp$-versions discontinuous Galerkin (DG) methods for the numerical solution of fractional subdiffusion problems of order $-α$ with $-1<α<0$ will be proposed and analyzed. Generic $hp$-version error estimates are derived after proving the stability of the approximate solution. For $h$-version DG approximations on appropriate graded meshes near$t=0$, we prove that the error is of order$O(k^{\max\{2,p\}+\fracα{2}})$, where $k$ is the maximum time-step size and $p\ge 1$ is the uniform degree of the DG solution. For $hp$-version DG approximations, by employing geometrically refined time-steps and linearly increasing approximation orders, exponential rates of convergence in the number of temporal degrees of freedom are shown. Finally, some numerical tests are given.

math.NA↗

arXiv2014

A discontinuous Petrov-Galerkin method for time-fractional diffusion equations

We propose and analyze a time-stepping discontinuous Petrov-Galerkin method combined with the continuous conforming finite element method in space for the numerical solution of time-fractional subdiffusion problems. We prove the existence, uniqueness and stability of approximate solutions, and derive error estimates. To achieve high order convergence rates from the time discretizations, the time mesh is graded appropriately near~$t=0$ to compensate the singular (temporal) behaviour of the exact solution near $t=0$ caused by the weakly singular kernel, but the spatial mesh is quasiuniform. In the $L_\infty((0,T);L_2(Ω))$-norm ($(0,T)$ is the time domain and $Ω$ is the spatial domain), for sufficiently graded time meshes, a global convergence of order $k^{m+α/2}+h^{r+1}$ is shown, where $0<α<1$ is the fractional exponent, $k$ is the maximum time step, $h$ is the maximum diameter of the spatial finite elements, and $m$ and $r$ are the degrees of approximate solutions in time and spatial variables, respectively. Numerical experiments indicate that our theoretical error bound is pessimistic. We observe that the error is of order ~$k^{m+1}+h^{r+1}$, that is, optimal in both variables.

math.NA↗

arXiv2014

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.

math.NA↗

arXiv2013

A Superconvergent discontinuous Galerkin method for Volterra integro-differential equations, smooth and non-smooth kernels

We study the numerical solution for Volerra integro-differential equations with smooth and non-smooth kernels. We use a $h$-version discontinuous Galerkin (DG) method and derive nodal error bounds that are explicit in the parameters of interest. In the case of non-smooth kernel, it is justified that the start-up singularities can be resolved at superconvergence rates by using non-uniformly graded meshes. Our theoretical results are numerically validated in a sample of test problems.

math.NA↗

arXiv2012

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

We consider an initial-boundary value problem for $\partial_tu-\partial_t^{-α}\nabla^2u=f(t)$, that is, for a fractional diffusion ($-1<α<0$) or wave ($0<α<1$) 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 $k^{2+α_-}+h^2\ell(k)$, uniformly in $t$, where $k$ is the maximum time step, $h$ is the maximum diameter of the spatial finite elements, $α_-=\min(α,0)\le0$ and $\ell(k)=\max(1,|\log k|)$. Here, we generalize a known result for the classical heat equation (i.e., the case $α=0$) by showing that at each time level $t_n$ the solution is superconvergent with respect to $k$: the error is of order $(k^{3+2α_-}+h^2)\ell(k)$. 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 $α<0$. 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 $k^{3+α_-}+h^2$.

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