Kassem Mustapha

Bernardo Cockburn, Kassem Mustapha

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 $\sqrt{\log(T h^{-2/(α+1)})}\, \,h^{k+2}$.

Kassem Mustapha, William McLean

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 $k^{3+α_-}+h^2$.

Kassem Mustapha, Basheer Abdallah, Khaled Furati

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.

Kassem Mustapha

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.

Samir Karaa, Kassem Mustapha, Amiya K. Pani

In this article, the piecewise-linear finite element method (FEM) is applied to approximate the solution of time-fractional diffusion equations on bounded convex domains. Standard energy arguments do not provide satisfactory results for such a problem due to the low regularity of its exact solution. Using a delicate energy analysis, {\it a priori} optimal error bounds in $L^2(Ω)$-, $H^1(Ω)$-norms, and a quasi-optimal bound in $L^{\infty}(Ω)$-norm are derived for the semidiscrete FEM for cases with smooth and nonsmooth initial data. The main tool of our analysis is based on a repeated use of an integral operator and use of a $t^m$ type of weights to take care of the singular behavior of the continuous solution at $t=0.$ The generalized Leibniz formula for fractional derivatives is found to play a key role in our analysis. Numerical experiments are presented to illustrate some of the theoretical results.

Kassem Mustapha

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.

Kim Ngan Le, William McLean, Kassem Mustapha

We present a new stability and convergence analysis for the spatial discretization of a time-fractional Fokker--Planck equation in a convex polyhedral domain, using continuous, piecewise-linear, finite elements. The forcing may depend on time as well as on the spatial variables, and the initial data may have low regularity. Our analysis uses a novel sequence of energy arguments in combination with a generalized Gronwall inequality. Although this theory covers only the spatial discretization, we present numerical experiments with a fully discrete scheme employing a very small time step, and observe results consistent with the predicted convergence behavior.

Josef Dick, Seungchan Ko, Quoc Thong Le Gia et al.

Due to divergence instability, the accuracy of low-order conforming finite element methods for nearly incompressible elasticity equations deteriorates as the Lamé coefficient $λ\to\infty$, or equivalently as the Poisson ratio $ν\to1/2$. This phenomenon, known as locking or non-robustness, remains not fully understood despite extensive investigation. In this work, we illustrate first that an analogous instability arises when applying the popular Physics-Informed Neural Networks (PINNs) to nearly incompressible elasticity problems, leading to significant loss of accuracy and convergence difficulties. Then, to overcome this challenge, we propose a robust decomposition-based PINN framework that reformulates the elasticity equations into balanced subsystems, thereby eliminating the ill-conditioning that causes locking. Our approach simultaneously solves the forward and inverse problems to recover both the decomposed field variables and the associated external conditions. We will also perform a convergence analysis to further enhance the reliability of the proposed approach. Moreover, through various numerical experiments, including constant, variable and parametric Lamé coefficients, we illustrate the efficiency of the proposed methodology.

Samir Karaa, Kassem Mustapha, Amiya K. Pani

In this paper, a semi-discrete spatial finite volume (FV) method is proposed and analyzed for approximating solutions of anomalous subdiffusion equations involving a temporal fractional derivative of order $α\in (0,1)$ in a two-dimensional convex polygonal domain. Optimal error estimates in $L^\infty(L^2)$- norm is shown to hold. Superconvergence result is proved and as a consequence, it is established that quasi-optimal order of convergence in $L^{\infty}(L^{\infty})$ holds. We also consider a fully discrete scheme that employs FV method in space, and a piecewise linear discontinuous Galerkin method to discretize in temporal direction. It is, further, shown that convergence rate is of order $O(h^2+k^{1+α}),$ where $h$ denotes the space discretizing parameter and $k$ represents the temporal discretizing parameter. Numerical experiments indicate optimal convergence rates in both time and space, and also illustrate that the imposed regularity assumptions are pessimistic.

Kim Ngan Le, William McLean, Kassem Mustapha

We study two schemes for a time-fractional Fokker-Planck equation with space- and time-dependent forcing in one space dimension. The first scheme is continuous in time and is discretized in space using a piecewise-linear Galerkin finite element method. The second is continuous in space and employs a time-stepping procedure similar to the classical implicit Euler method. We show that the space discretization is second-order accurate in the spatial $L_2$-norm, uniformly in time, whereas the corresponding error for the time-stepping scheme is $O(k^α)$ for a uniform time step $k$, where $α\in(1/2,1)$ is the fractional diffusion parameter. In numerical experiments using a combined, fully-discrete method, we observe convergence behaviour consistent with these results.

Kassem Mustapha, Maher Nour, Bernardo Cockburn

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.