scholarly journals NUMERICAL SOLVING UNSTEADY SPACE-FRACTIONAL PROBLEMS WITH THE SQUARE ROOT OF AN ELLIPTIC OPERATOR

2016 ◽  
Vol 21 (2) ◽  
pp. 220-238 ◽  
Author(s):  
Petr N. Vabishchevich
Keyword(s):  
Elliptic Operator ◽  
Finite Element Approximation ◽  
Second Order ◽  
Evolutionary Equation ◽  
Element Approximation ◽  
Square Root ◽  
Dimensional Problem ◽  
Order Accuracy ◽  
First Order ◽  
Fractional Equation

An unsteady problem is considered for a space-fractional equation in a bounded domain. A first-order evolutionary equation involves the square root of an elliptic operator of second order. Finite element approximation in space is employed. To construct approximation in time, regularized two- level schemes are used. The numerical implementation is based on solving the equation with the square root of the elliptic operator using an auxiliary Cauchy problem for a pseudo-parabolic equation. The scheme of the second-order accuracy in time is based on a regularization of the three-level explicit Adams scheme. More general problems for the equation with convective terms are considered, too. The results of numerical experiments are presented for a model two-dimensional problem.

scholarly journals Numerical Solution of Time-Dependent Problems with Fractional Power Elliptic Operator

2018 ◽  
Vol 18 (1) ◽  
pp. 111-128 ◽  
Author(s):  
Petr N. Vabishchevich
Keyword(s):  
Elliptic Operator ◽  
Fractional Power ◽  
Finite Element Approximation ◽  
Evolutionary Equation ◽  
Quadrature Formulas ◽  
Element Approximation ◽  
Discrete Problem ◽  
Time Standard ◽  
Time Dependent Problems ◽  
Fractional Equation

AbstractAn unsteady problem is considered for a space-fractional equation in a bounded domain. A first-order evolutionary equation involves a fractional power of an elliptic operator of second order. Finite element approximation in space is employed. To construct approximation in time, standard two-level schemes are used. The approximate solution at a new time-level is obtained as a solution of a discrete problem with the fractional power of the elliptic operator. A Padé-type approximation is constructed on the basis of special quadrature formulas for an integral representation of the fractional power elliptic operator using explicit schemes. A similar approach is applied in the numerical implementation of implicit schemes. The results of numerical experiments are presented for a test two-dimensional problem.

scholarly journals Numerical Solution of Nonstationary Problems for a Space-Fractional Diffusion Equation

2016 ◽  
Vol 19 (1) ◽  
Author(s):  
Petr N. Vabishchevich
Keyword(s):  
Diffusion Equation ◽  
Elliptic Operator ◽  
Fractional Power ◽  
Finite Element Approximation ◽  
Fractional Diffusion ◽  
Fractional Diffusion Equation ◽  
Evolutionary Equation ◽  
Element Approximation ◽  
First Order ◽  
Space Fractional Diffusion Equation

AbstractAn unsteady problem is considered for a space-fractional diffusion equation in abounded domain. A first-order evolutionary equation containing a fractional power of an elliptic operator of second order is studied for general boundary conditions of Robin type. Finite element approximation in space is employed. To construct approximation in time, regularized two-level schemes are used. The numerical implementation is based on solving the equation with the fractional power of the elliptic operator using an auxiliary Cauchy problem for a pseudo-parabolic equation. The results of numerical experiments are presented for a model two-dimensional problem.

scholarly journals Numerical approximation of fractional powers of elliptic operators

2019 ◽  
Vol 40 (3) ◽  
pp. 1746-1771 ◽  
Author(s):  
Beiping Duan ◽  
Raytcho D Lazarov ◽  
Joseph E Pasciak
Keyword(s):  
Finite Element ◽  
Elliptic Boundary ◽  
Finite Element Approximation ◽  
Elliptic Operators ◽  
Second Order ◽  
Element Approximation ◽  
Fractional Powers ◽  
Time Stepping ◽  
Order Elliptic Operator ◽  
Graded Meshes

Abstract In this paper, we develop and study algorithms for approximately solving linear algebraic systems: ${{\mathcal{A}}}_h^\alpha u_h = f_h$, $ 0< \alpha <1$, for $u_h, f_h \in V_h$ with $V_h$ a finite element approximation space. Such problems arise in finite element or finite difference approximations of the problem $ {{\mathcal{A}}}^\alpha u=f$ with ${{\mathcal{A}}}$, for example, coming from a second-order elliptic operator with homogeneous boundary conditions. The algorithms are motivated by the method of Vabishchevich (2015, Numerically solving an equation for fractional powers of elliptic operators. J. Comput. Phys., 282, 289–302) that relates the algebraic problem to a solution of a time-dependent initial value problem on the interval $[0,1]$. Here we develop and study two time-stepping schemes based on diagonal Padé approximation to $(1+x)^{-\alpha }$. The first one uses geometrically graded meshes in order to compensate for the singular behaviour of the solution for $t$ close to $0$. The second algorithm uses uniform time stepping, but requires smoothness of the data $f_h$ in discrete norms. For both methods, we estimate the error in terms of the number of time steps, with the regularity of $f_h$ playing a major role for the second method. Finally, we present numerical experiments for ${{\mathcal{A}}}_h$ coming from the finite element approximations of second-order elliptic boundary value problems in one and two spatial dimensions.

scholarly journals FINITE ELEMENT APPROXIMATION OF A TIME-FRACTIONAL DIFFUSION PROBLEM FOR A DOMAIN WITH A RE-ENTRANT CORNER

2017 ◽  
Vol 59 (1) ◽  
pp. 61-82 ◽  
Author(s):  
KIM NGAN LE ◽  
WILLIAM MCLEAN ◽  
BISHNU LAMICHHANE
Keyword(s):  
Piecewise Linear ◽  
Finite Element Approximation ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Fractional Diffusion ◽  
Diffusion Problem ◽  
Second Order ◽  
Element Approximation ◽  
Second Order Convergence ◽  
Initial Boundary Value

An initial-boundary value problem for a time-fractional diffusion equation is discretized in space, using continuous piecewise-linear finite elements on a domain with a re-entrant corner. Known error bounds for the case of a convex domain break down, because the associated Poisson equation is no longer $H^{2}$-regular. In particular, the method is no longer second-order accurate if quasi-uniform triangulations are used. We prove that a suitable local mesh refinement about the re-entrant corner restores second-order convergence. In this way, we generalize known results for the classical heat equation.

The Special Form of Second Order Accuracy N-S Equations for Incompressible Fluid

2014 ◽  
Vol 644-650 ◽  
pp. 1644-1647
Author(s):  
Zhan Song Li ◽  
Shi Jiang Zhu
Keyword(s):  
Laminar Flow ◽  
Turbulent Flow ◽  
Incompressible Fluid ◽  
Special Form ◽  
Higher Order ◽  
Second Order ◽  
Time And Space ◽  
Order Accuracy ◽  
First Order

Classic N-S equation has first order accuracy in both of time and space. It has only the terms of first order, without the terms of second or higher order. These terms are relative in time and space steps. The time and space steps, as basic elements of fluid research, should be only some finite quantities and not be infinitely near to zero as defined in mathematics. If the terms of second or higher order can be ignored depends on the value of the corresponding derivative multiplied. Compared with terms of first order, the terms of second or higher order can be ignored under the condition of laminar flow. However, under the condition of turbulent flow, these can’t be ignored yet. When turbulent flow develops fully, the terms of first order, compared with terms of second order, can be ignored. So, it is why classic N-S equations aren’t closed when they are used to analyze turbulent flow. On the basic, many different special forms of the second order accuracy N-S equations of incompressible fluid are derived.

A New Single Step Full Discrete Scheme for a Semilinear Second Order Hyperbolic Equation

2011 ◽  
Vol 314-316 ◽  
pp. 667-671
Author(s):  
Kang Deng
Keyword(s):  
Finite Element Approximation ◽  
Single Step ◽  
Second Order ◽  
Element Approximation ◽  
Initial Value ◽  
Discrete Scheme ◽  
Fully Discrete ◽  
Order Hyperbolic Equation ◽  
Discrete Finite Element ◽  
Second Order Hyperbolic Equation

In this paper, we study a simplified single step full discrete scheme for a class of semilinear hyperbolic problems of second order. At first we obtain a system of second ordinary differential equations with initial value by use of spatially discrete finite element approximation with interpolated coefficients. Next in terms of a single step scheme to the time variable for this system we gain a fully discrete scheme with high accuracy. Finally we give the stable and convergence of the full discrete schemes.

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