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We consider the time discretization of a linear parabolic problem by the discontinuous Galerkin (DG) method using piecewise polynomials of degree at most r − 1 in t, for r ≥ 1 and with maximum step size k. It is well known that the... more
We consider the time discretization of a linear parabolic problem by the discontinuous Galerkin (DG) method using piecewise polynomials of degree at most r − 1 in t, for r ≥ 1 and with maximum step size k. It is well known that the spatial L2-norm of the DG error is of optimal order kr globally in time, and is, for r ≥ 2, superconvergent of order k2r− 1 at the nodes. We show that on the n th subinterval (tn− 1,tn), the dominant term in the DG error is proportional to the local right Radau polynomial of degree r. This error profile implies that the DG error is of order kr+ 1 at the right-hand Gauss–Radau quadrature points in each interval. We show that the norm of the jump in the DG solution at the left end point tn− 1 provides an accurate a posteriori estimate for the maximum error over the subinterval (tn− 1,tn). Furthermore, a simple post-processing step yields a continuous piecewise polynomial of degree r with the optimal global convergence rate of order kr+ 1. We illustrate thes...
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A second-order accurate time-stepping scheme for solving a time-fractional Fokker–Planck equation of order $\alpha \in (0, 1)$, with a general driving force, is investigated. A stability bound for the semidiscrete solution is obtained for... more
A second-order accurate time-stepping scheme for solving a time-fractional Fokker–Planck equation of order $\alpha \in (0, 1)$, with a general driving force, is investigated. A stability bound for the semidiscrete solution is obtained for $\alpha \in (1/2,1)$ via a novel and concise approach. Our stability estimate is $\alpha $-robust in the sense that it remains valid in the limiting case where $\alpha $ approaches $1$ (when the model reduces to the classical Fokker–Planck equation), a limit that presents practical importance. Concerning the error analysis, we obtain an optimal second-order accurate estimate for $\alpha \in (1/2,1)$. A time-graded mesh is used to compensate for the singular behavior of the continuous solution near the origin. The time-stepping scheme scheme is associated with a standard spatial Galerkin finite element discretization to numerically support our theoretical contributions. We employ the resulting fully discrete computable numerical scheme to perform so...
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We establish the well-posedness of an initial-boundary value problem for a general class of linear time-fractional, advection-diffusion-reaction equations, allowing space- and time-dependent coefficients as well as initial data that may... more
We establish the well-posedness of an initial-boundary value problem for a general class of linear time-fractional, advection-diffusion-reaction equations, allowing space- and time-dependent coefficients as well as initial data that may have low regularity. Our analysis relies on novel energy methods in combination with a fractional Gronwall inequality and properties of fractional integrals.
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Research Interests: Mathematics, Applied Mathematics, Computer Science, Theory Of Computation, Finite Element, and 9 moreMathematical Analysis, Numerical Algorithms, Piecewise Linear, Discretization, Fractional Calculus, Numerical Analysis and Computational Mathematics, Convergence Analysis, Galerkin Method, and Heat Equation
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K. MUSTAPHA School of Mathematics, University of New South Wales Sydney 2052, Australia kassim©maths, unsw. edu. au Abstract--In this work, we analyse the error of a discrete Petrov-Galerkin scheme for nonlin- ear ruth-order ordinary... more
K. MUSTAPHA School of Mathematics, University of New South Wales Sydney 2052, Australia kassim©maths, unsw. edu. au Abstract--In this work, we analyse the error of a discrete Petrov-Galerkin scheme for nonlin- ear ruth-order ordinary differential and integrodifferential equations on a finite interval subject to nonlinear side conditions. As a trial space we chose high-order Cm-splines. We prove optimal-order convergence and superconvergence in the knots for lower-order derivatives, where the range of deriva- tives for these enhanced convergences to hold is determined by the behaviour of the nonlocal part of the integrodifferential equation. Our results extend and simplify earlier results by Ganesh and Sloan [1]. The numerical experiments in this work, for several singularly perturbed ordinary differential equa- tions, demonstrate the power of our scheme that does not require any mesh restriction. (~) 2005 Elsevier Ltd. All rights reserved. Keywords--Discrete Petrov-Galerkin method, ...
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In this paper, a weakly dissipative viscoelastic plate equation with an infinite memory is considered. We show a general energy decay rate for a wide class of relaxation functions. To support our theoretical findings, some numerical... more
In this paper, a weakly dissipative viscoelastic plate equation with an infinite memory is considered. We show a general energy decay rate for a wide class of relaxation functions. To support our theoretical findings, some numerical illustrations are presented at the end. The numerical solution is computed using the popular finite element method in space, combined with time-stepping finite differences.
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Abstract We propose and analyze a fully discrete Laplace modified alternating direction implicit quadrature PetrovGalerkin (ADI-QPG) method for solving parabolic initial-boundary value problems on rectangular domains. We prove optimal... more
Abstract We propose and analyze a fully discrete Laplace modified alternating direction implicit quadrature PetrovGalerkin (ADI-QPG) method for solving parabolic initial-boundary value problems on rectangular domains. We prove optimal order convergence results for ...
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We propose and analyze an application of a fully discrete C2 spline quadrature Petrov‐Galerkin method for spatial discretization of semi‐linear parabolic initial‐boundary value problems on rectangular domains. We prove second order in... more
We propose and analyze an application of a fully discrete C2 spline quadrature Petrov‐Galerkin method for spatial discretization of semi‐linear parabolic initial‐boundary value problems on rectangular domains. We prove second order in time and optimal order H1 norm convergence in space for the extrapolated Crank‐Nicolson quadrature Petrov‐Galerkin scheme. We demonstrate numerically both L2 and H1 norm optimal order convergence of the scheme even if the nonlinear source term is not smooth. © 2005 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2005.
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We consider an initial value problem for a class of evolution equations incorporating a memory term with a weakly singular kernel bounded by C ( t − s ) α − 1 C(t-s)^{\alpha -1} , where 0 > α > 1 0>\alpha >1 . For the time... more
We consider an initial value problem for a class of evolution equations incorporating a memory term with a weakly singular kernel bounded by C ( t − s ) α − 1 C(t-s)^{\alpha -1} , where 0 > α > 1 0>\alpha >1 . For the time discretization we apply the discontinuous Galerkin method using piecewise polynomials of degree at most q − 1 q-1 , for q = 1 q=1 or 2 2 . For the space discretization we use continuous piecewise-linear finite elements. The discrete solution satisfies an error bound of order k q + h 2 ℓ ( k ) k^q+h^2\ell (k) , where k k and h h are the mesh sizes in time and space, respectively, and ℓ ( k ) = max ( 1 , log k − 1 ) \ell (k)=\max (1,\log k^{-1}) . In the case q = 2 q=2 , we prove a higher convergence rate of order k 3 + h 2 ℓ ( k ) k^3+h^2\ell (k) at the nodes of the time mesh. Typically, the partial derivatives of the exact solution are singular at t = 0 t=0 , necessitating the use of non-uniform time steps. We compare our theoretical error bounds...
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The permeability of a 3D geological fracture network is determined by triangulating the fractures and solving the 2D Darcy's equation in each fracture. Here, the numerical modelling... more
The permeability of a 3D geological fracture network is determined by triangulating the fractures and solving the 2D Darcy's equation in each fracture. Here, the numerical modelling aims to simulate a great number of networks made up of a great number of fractures i.e. from 10 to 10 fractures. Parallel computing allows us to solve very large linear systems improving
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ABSTRACT