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A Parareal exponential integrator finite element method for linear parabolic equations

Jianguo Huang, Yuejin Xu

TL;DR

This work addresses fast, accurate solution of linear parabolic PDEs on rectangular domains by marrying finite element spatial discretization with exponential time integrators and the Parareal parallel-in-time framework. The proposed PEIFE-linear method achieves unconditional stability for linear problems, provable $L^2$ error estimates, and a fast FFT-based implementation due to simultaneous diagonalization of mass and coefficient matrices. Theoretical results quantify the error in terms of time-step sizes and mesh size, and numerical experiments in 1D–3D validate convergence rates and demonstrate substantial parallel speedups. The approach extends naturally to periodic boundaries and offers a practical tool for large-scale parabolic simulations.

Abstract

In this paper, for solving a class of linear parabolic equations in rectangular domains, we have proposed an efficient Parareal exponential integrator finite element method. The proposed method first uses the finite element approximation with continuous multilinear rectangular basis function for spatial discretization, and then takes the Runge-Kutta approach accompanied with Parareal framework for time integration of the resulting semi-discrete system to produce parallel-in-time numerical solution. Under certain regularity assumptions, fully-discrete error estimates in $L^2$-norm are derived for the proposed schemes with random interpolation nodes. Moreover, a fast solver can be provided based on tensor product spectral decomposition and fast Fourier transform (FFT), since the mass and coefficient matrices of the proposed method can be simultaneously diagonalized with an orthogonal matrix. A series of numerical experiments in various dimensions are also presented to validate the theoretical results and demonstrate the excellent performance of the proposed method.

A Parareal exponential integrator finite element method for linear parabolic equations

TL;DR

This work addresses fast, accurate solution of linear parabolic PDEs on rectangular domains by marrying finite element spatial discretization with exponential time integrators and the Parareal parallel-in-time framework. The proposed PEIFE-linear method achieves unconditional stability for linear problems, provable error estimates, and a fast FFT-based implementation due to simultaneous diagonalization of mass and coefficient matrices. Theoretical results quantify the error in terms of time-step sizes and mesh size, and numerical experiments in 1D–3D validate convergence rates and demonstrate substantial parallel speedups. The approach extends naturally to periodic boundaries and offers a practical tool for large-scale parabolic simulations.

Abstract

In this paper, for solving a class of linear parabolic equations in rectangular domains, we have proposed an efficient Parareal exponential integrator finite element method. The proposed method first uses the finite element approximation with continuous multilinear rectangular basis function for spatial discretization, and then takes the Runge-Kutta approach accompanied with Parareal framework for time integration of the resulting semi-discrete system to produce parallel-in-time numerical solution. Under certain regularity assumptions, fully-discrete error estimates in -norm are derived for the proposed schemes with random interpolation nodes. Moreover, a fast solver can be provided based on tensor product spectral decomposition and fast Fourier transform (FFT), since the mass and coefficient matrices of the proposed method can be simultaneously diagonalized with an orthogonal matrix. A series of numerical experiments in various dimensions are also presented to validate the theoretical results and demonstrate the excellent performance of the proposed method.

Paper Structure

This paper contains 10 sections, 7 theorems, 57 equations, 3 figures, 7 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. The EIFE-linear method and PEIFE-linear method
  3. Semi-discretization in space by finite element approximation
  4. Linear exponential integrator finite element method
  5. Exponential integrator finite element method in parallel-in-time pattern
  6. Convergence analysis of EIFE-linear method
  7. Convergence analysis of PEIFE-linear method for linear problems
  8. Numerical experiments and applications
  9. Convergence tests
  10. Conclusions

Key Result

Lemma 3.1

Let $u_h(t)$ and $u(t)$ be the solutions of eq1-5 and eq3-1. Assume that $u(t)$ fulfills Assumption assumption3. Then the error bound becomes

Figures (3)

  • Figure 1: The evolutions of $L^2$-error and $L^{\infty}$-error declining curves of the numerical solution produced by PEIFE-linear22, PEIFE-linear23 and PEIFE-linear33 schemes for Example \ref{['ex6']}.
  • Figure 2: The evolutions of $L^2$-error and $L^{\infty}$-error declining curves of the numerical solution produced by PEIFE-linear22, PEIFE-linear23 and PEIFE-linear33 schemes for Example \ref{['ex1']}.
  • Figure 3: Numerical results at $t=0,1.25,0.5,0.75,1$ produced by PEIFE-linear22, PEIFE-linear23, PEIFE-linear33 schemes (from left to right) for Example \ref{['ex7']} with various $\alpha$ and $f$.

Theorems & Definitions (19)

  • Lemma 3.1
  • Theorem 3.1
  • proof
  • Remark 3.1
  • Lemma 4.1
  • Theorem 4.1
  • proof
  • Theorem 4.2
  • proof
  • Remark 4.1
  • ...and 9 more