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In this paper, a finite volume element (FVE) method is considered for spatial approximations of time fractional diffusion equations involving a Riemann-Liouville fractional derivative of order α∈ (0, 1) in time. Improving upon earlier results [Karaa et al., IMA J. Numer. Anal.37(2017) 945–964], error estimates in L2(Ω)- and H1(Ω)-norms for the semidiscrete problem with smooth and mildly smooth initial data, i.e., v∈ H2(Ω) ∩ H01(Ω) and v∈ H01(Ω) are established. For nonsmooth data, that is, v∈  L2(Ω), the optimal L2(Ω)-error estimate is shown to hold only under an additional assumption on the triangulation, which is known to be satisfied for symmetric triangulations. Superconvergence result is also proved and as a consequence, a quasi-optimal error estimate is established in the L∞(Ω)-norm. Further, two fully discrete schemes using convolution quadrature in time generated by the backward Euler and the second-order backward difference methods are analyzed, and error estimates are derived for both smooth and nonsmooth initial data. Based on a comparison of the standard Galerkin finite element solution with the FVE solution and exploiting tools for Laplace transforms with semigroup type properties of the FVE solution operator, our analysis is then extended in a unified manner to several time fractional order evolution problems. Finally, several numerical experiments are conducted to confirm our theoretical findings.

The paper is concerned with the strong approximation of a stochastic time-fractional Allen-Cahn equation driven by an additive fractionally integrated Gaussian noise. The model involves two nonlocal terms in time; namely, a Caputo fractional derivative of order α ∈ (0 , 1) , and a Riemann–Liouville fractional integral operator of order γ ∈ [ 0 , 1 ] applied to a Gaussian noise. We approximate the model by a standard piecewise linear finite element method (FEM) in space and the classical Grünwald–Letnikov method in time (for both time-fractional operators), and the noise by the L 2 -projection. Spatially semidiscrete and fully discrete schemes are analyzed and strong convergence rates are obtained by exploiting the temporal Hölder continuity property of the solution. Numerical experiments are presented to illustrate the theoretical results. [ABSTRACT FROM AUTHOR]

This article develops a new two‐level three‐point implicit finite difference scheme of order 2 in time and 4 in space based on arithmetic average discretization for the solution of nonlinear parabolic equation ε uxx= f(x, t, u, ux, ut), 0 < x< 1, t> 0 subject to appropriate initial and Dirichlet boundary conditions, where ε > 0 is a small positive constant. We also propose a new explicit difference scheme of order 2 in time and 4 in space for the estimates of (∂u/∂x). The main objective is the proposed formulas are directly applicable to both singular and nonsingular problems. We do not require any fictitious points outside the solution region and any special technique to handle the singular problems. Stability analysis of a model problem is discussed. Numerical results are provided to validate the usefulness of the proposed formulas. © 2006 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2007

We propose a 9‐point fourth‐order finite difference scheme for 2D elliptic problems with a mixed derivative and variable coefficients. The same approach is extended to derive a class of two‐level high‐order compact schemes with weighted time discretization for solving 2D parabolic problems with a mixed derivative. The schemes are fourth‐order accurate in space and second‐ or lower‐order accurate in time depending on the choice of a weighted average parameter μ. Unconditional stability is proved for 0.5 ≤ μ ≤ 1, and numerical experiments supporting our theoretical analysis and confirming the high‐order accuracy of the schemes are presented. © 2006 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 23: 366–378, 2007

We establish upper bounds for the eigenvalues of second-order and fourth-order differential equations. The inequalities are obtained via rearrangements of higher degree.

We derive a fourth‐order finite difference scheme for the two‐dimensional convection‐diffusion equation on an hexagonal grid. The difference scheme is defined on a single regular hexagon of size hover a seven‐point stencil. Numerical experiments are conducted to verify the high accuracy of the derived scheme, and to compare it with the standard second‐order central difference scheme. © 2006 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2006

We derive a high‐order compact alternating direction implicit (ADI) method for solving three‐dimentional unsteady convection‐diffusion problems. The method is fourth‐order in space and second‐order in time. It permits multiple uses of the one‐dimensional tridiagonal algorithm with a considerable saving in computing time and results in a very efficient solver. It is shown through a discrete Fourier analysis that the method is unconditionally stable in the diffusion case. Numerical experiments are conducted to test its high order and to compare it with the standard second‐order Douglas‐Gunn ADI method and the spatial fourth‐order compact scheme by Karaa. © 2005 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2006

Abstract: We develop numerical methods for the computer simulation and modeling of a three dimensional heat transfer problem in biological bodies. The technique is intended for the temperature predications and parameter measurements in thermal medical practices and for the studies of thermomechanical interaction of biological bodies at high temperature. We examine a mathematical model based on the classical well-known Pennes equation for heat transfer in biological bodies. A finite difference discretization scheme is used to discretize the governing partial differential equation. A preconditioned iterative solver is employed to solve the resulting sparse linear system at each time step. Numerical results are obtained to demonstrate the efficacy of the proposed numerical methods. [Copyright &y& Elsevier]

On e´tudie le proble`me de maximisation des valeurs propres de l'e´quation (q(x)y)´$´+ λρ(x)y = 0 de´finie sur un intervalle borne´ de R . Le proble`me est re´solu a` l'aide de conditions suffisantes d'optimalite´.