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A Posteriori Error Estimates for Fully Discrete Finite Element Method for Generalized Diffusion Equation with Delay.

Wansheng Wang1,2, Lijun Yi1, Aiguo Xiao3

  • 1Department of Mathematics, Shanghai Normal University, Shanghai, 200234 China.

Journal of Scientific Computing
|August 25, 2020
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Summary
This summary is machine-generated.

This study introduces novel a posteriori error estimators for generalized diffusion equations with delays. These estimators provide the first rigorous upper and lower bounds for fully discrete approximations, confirmed by numerical experiments.

Keywords:
A posteriori error estimatesCrank–Nicolson methodFinite element methodGeneralized diffusion equation with delayLong-time a posteriori error estimates

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Area of Science:

  • Numerical analysis
  • Partial differential equations
  • Computational mathematics

Background:

  • Generalized diffusion equations with delays present significant challenges in numerical analysis.
  • Accurate error estimation is crucial for reliable simulations of such systems.
  • Existing methods often lack rigorous bounds for fully discrete approximations.

Purpose of the Study:

  • To derive novel a posteriori error estimators for generalized diffusion equations with delay.
  • To establish the first a posteriori upper and lower error bounds for fully discrete approximations.
  • To obtain long-time a posteriori error estimates for stable systems.

Main Methods:

  • Utilizing the Crank-Nicolson method for time discretization.
  • Employing a continuous, piecewise linear finite element space for spatial discretization.
  • Applying interpolation estimates and piecewise quadratic reconstructions for error analysis.
  • Using linear approximations for the delay term to estimate approximation errors.

Main Results:

  • Development of several a posteriori error estimators for the generalized diffusion equation with delay.
  • Derivation of the first a posteriori upper and lower error bounds for fully discrete approximations.
  • Obtained long-time a posteriori error estimates for stable systems.
  • Numerical experiments validated the theoretical findings.

Conclusions:

  • The derived a posteriori error estimators provide reliable bounds for numerical solutions.
  • This work advances the understanding and simulation of diffusion equations with delays.
  • The findings are particularly relevant for long-time simulations of stable systems.