Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

High order numerical schemes for solving fractional powers of elliptic operators

Raimondas Ciegis, Petr Vabishchevich

TL;DR

The paper tackles numerical approximation of fractional power elliptic problems $\mathcal{A}^\alpha u=f$ with $0<\alpha<1$ by converting them into a Cauchy problem for a pseudo-parabolic equation and solving it with finite element spatial discretization. It develops a family of high-order three-level schemes, achieving fourth-order accuracy through an optimal weight parameter $\sigma_0 = \frac{2+\alpha}{6\alpha}$ while maintaining unconditional stability under suitable conditions. The approach relies on a robust time-stepping framework for the auxiliary problem and careful initialization of the first time level; stability and convergence are supported by theoretical analysis and 2D numerical experiments with discontinuous coefficients. The results demonstrate that high-order schemes and properly prepared initial data significantly improve accuracy, with graded time grids offering a path to improved convergence for less regular solutions.

Abstract

In many recent applications when new materials and technologies are developed it is important to describe and simulate new nonlinear and nonlocal diffusion transport processes. A general class of such models deals with nonlocal fractional power elliptic operators. In order to solve these problems numerically it is proposed (Petr N. Vabishchevich, Journal of Computational Physics. 2015, Vol. 282, No.1, pp.289--302) to consider equivalent local nonstationary initial value pseudo-parabolic problems. Previously such problems were solved by using the standard implicit backward and symmetrical Euler methods. In this paper we use the one-parameter family of three-level finite difference schemes for solving the initial value problem for the first order nonstationary pseudo-parabolic problem. The fourth-order approximation scheme is developed by selecting the optimal value of the weight parameter. The results of the theoretical analysis are supplemented by results of extensive computational experiments.

High order numerical schemes for solving fractional powers of elliptic operators

TL;DR

The paper tackles numerical approximation of fractional power elliptic problems with by converting them into a Cauchy problem for a pseudo-parabolic equation and solving it with finite element spatial discretization. It develops a family of high-order three-level schemes, achieving fourth-order accuracy through an optimal weight parameter while maintaining unconditional stability under suitable conditions. The approach relies on a robust time-stepping framework for the auxiliary problem and careful initialization of the first time level; stability and convergence are supported by theoretical analysis and 2D numerical experiments with discontinuous coefficients. The results demonstrate that high-order schemes and properly prepared initial data significantly improve accuracy, with graded time grids offering a path to improved convergence for less regular solutions.

Abstract

In many recent applications when new materials and technologies are developed it is important to describe and simulate new nonlinear and nonlocal diffusion transport processes. A general class of such models deals with nonlocal fractional power elliptic operators. In order to solve these problems numerically it is proposed (Petr N. Vabishchevich, Journal of Computational Physics. 2015, Vol. 282, No.1, pp.289--302) to consider equivalent local nonstationary initial value pseudo-parabolic problems. Previously such problems were solved by using the standard implicit backward and symmetrical Euler methods. In this paper we use the one-parameter family of three-level finite difference schemes for solving the initial value problem for the first order nonstationary pseudo-parabolic problem. The fourth-order approximation scheme is developed by selecting the optimal value of the weight parameter. The results of the theoretical analysis are supplemented by results of extensive computational experiments.

Paper Structure

This paper contains 6 sections, 3 theorems, 63 equations, 10 figures.

Table of Contents

  1. Introduction
  2. Problem Formulation
  3. Cauchy problem method
  4. Three level schemes
  5. Numerical Experiments
  6. Conclusions

Key Result

Theorem 1

For $\sigma \geq 0.5$ the difference scheme 12, 13 is unconditionally stable with respect to the initial data. The approximate solution satisfies the estimate

Figures (10)

  • Figure 1: The reference solution $\bar{w}_N$: top -- $\alpha = 0.25$, center -- $\alpha = 0.5$, bottom -- $\alpha = 0.75$
  • Figure 2: The accuracy of the two-level scheme \ref{['12']} for $\alpha = 0.25$: top -- $L_2(\Omega)$, bottom -- $H^1(\Omega)$
  • Figure 3: The accuracy of the two-level scheme \ref{['12']} for $\alpha = 0.5$: top -- $L_2(\Omega)$, bottom -- $H^1(\Omega)$
  • Figure 4: The accuracy of the two-level scheme \ref{['12']} for $\alpha = 0.75$: top -- $L_2(\Omega)$, bottom -- $H^1(\Omega)$
  • Figure 5: The accuracy of the three-level scheme \ref{['15']} with the initial condition computed using the standard two-level scheme \ref{['21']} for $\alpha = 0.25$ ($\sigma_0 = 3/2$): top -- $L_2(\Omega)$, bottom -- $H^1(\Omega)$.
  • ...and 5 more figures

Theorems & Definitions (3)

  • Theorem 1
  • Theorem 2
  • Theorem 3