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Extending a new two-grid waveform relaxation on a spatial finite element discretization

A. Mesforush, N. Habibi

arXiv:2609.27935v1math.NA

TL;DR

The paper extends a two-grid method to time-dependent PDEs, using finite elements and waveform relaxation to develop a new two-grid scheme and analyze its convergence. In tests on the two-dimensional heat equation, the new scheme has a better averaged convergence factor than Gauss-Seidel and classical two-grid methods.

  • Problem

    The paper addresses extending an efficient two-grid method for elliptic PDEs to time-dependent equations.

    Source: §1 · p. 1

  • Method

    The method of lines with finite-element spatial discretization yields an ODE system solved by a new two-grid waveform relaxation scheme, whose convergence is analyzed through its operator's spectral radius.

    Source: §2 · p. 5, §4 · pp. 14–15

  • Results

    The new two-grid waveform relaxation method has a better averaged convergence factor than Gauss-Seidel and classical two-grid methods.

    Source: §4 · p. 15

  • Takeaways & Limitations

    The method is presented for time-dependent equations, with its performance and analysis tested on the two-dimensional heat equation.

    Source: §4 · pp. 14–15

    The study treats the continuous-time case; the discontinuous-time case is left for future work.

    Source: §4 · p. 15

Abstract

from arXiv · show

In this work, a new two-grid method presented for the elliptic partial differential equations is generalized to the time-dependent linear parabolic partial differential equations. The new two-grid waveform relaxation method uses the numerical method of lines, replacing any spatial derivative by a discrete formula, obtained here by the finite element method. A convergence analysis in terms of the spectral radius of the corresponding two-grid waveform relaxation operator is also developed. Moreover, the efficiency of the presented method and its analysis are tested, applying the two-dimensional heat equation.

1. Introduction

The introduction motivates extending a two-grid method for the Poisson equation to time-dependent PDEs. It outlines a finite-element waveform-relaxation approach using both classical and new two-grid schemes.

1. Introduction

  • The new two-grid waveform relaxation method extends the Poisson-equation approach of Moghaderi and Dehghan to time-dependent equations.The method uses nested spatial grids to produce each new iterate from the previous one.Source: §1 · pp. 1, 4
  • The model problem is the two-dimensional heat equation, discretized in space with finite elements to obtain a system of ODEs.The setup uses continuous piecewise linear functions on an admissible triangulation.Source: §1 · p. 2
  • Waveform relaxation applies directly to the semidiscrete ODE system and works parallel in time.The method splits the mass and stiffness matrices; the described smoother uses Gauss-Seidel splittings.Source: §1 · pp. 2–3
  • The classical two-grid waveform relaxation scheme uses presmoothing, coarse-grid correction, and postsmoothing on nested spatial grids.Its described transfer operators use linear interpolation and adjoint restriction; the heat-equation setup uses full weighting and bilinear interpolation.Source: §1 · pp. 3–4
  • The new scheme applies Gauss-Seidel iterations, computes a two-grid correction, blends it with an intermediate iterate, and applies a final two-grid solve.The intermediate iterate is formed by adding a scaled correction to the result of the initial Gauss-Seidel steps.Source: §1 · pp. 4–5
  • The paper develops spectral-radius convergence analysis and compares Gauss-Seidel, classical two-grid, and new two-grid methods using averaged convergence factors.The analysis uses Volterra integral equations and a Laplace transform to handle the time-dependent problem.Source: §1 · p. 5

2. convergence analysis

The convergence analysis uses Volterra integral-equation theory and Laplace transformation to study the two-grid waveform-relaxation operators through spectral radii. It extends the classical-cycle analysis to the new cycle for finite and infinite time intervals.

2.1. Classical two-grid convergence.

  • The classical two-grid cycle is represented by an operator M, with finite- and infinite-time spectral-radius results extended to the new method.The analysis treats the classical cycle as a successive-approximation method before extending the results to NTGW.Source: §2.1 · pp. 6–7, §2.2 · p. 7

2.2. New two-grid convergence.

  • The NTGW cycle is written as u^(ν)=Nu^(ν−1)+ψ, with N formed from products of the standard waveform-relaxation and new correction operators.N_c is a linear combination of products of K_c and CR_c.Source: §2.2 · pp. 7, 8, 10
  • For finite intervals, the new operator N is bounded, and its spectral-radius expression follows from the continuous-kernel result on [0,T].The proof uses continuity of the K_c and CR_c operators and the resulting kernel of N_c.Source: §2.2 · p. 11
  • For infinite intervals, N is bounded when the stated eigenvalues have positive real parts, and its kernel belongs to L1(0,∞).The L1 kernel result permits application of the infinite-interval spectral-radius lemma.Source: §2.2 · pp. 11–12

3. Numerical results

The numerical study compares Gauss-Seidel waveform relaxation, classical two-grid waveform relaxation, and the new two-grid method on two-dimensional heat-equation examples. The authors describe the new method's results as satisfactory in both examples.

3. Numerical results

  • The study selects NTGW's ω by scanning candidate values and computing the infinity norm of the error; it reports optimum values for multiple spatial discretizations.Figure 1 shows the parameter on a 49 × 49 mesh with ν1 = ν2 = ν3 = 1.Source: §3 · p. 13, Figure 1 · p. 13, Table 1 · p. 14
  • With h = 2^-7, the first example varies the Crank–Nicolson time step from 0.04 to 0.001 and compares the three methods' convergence factors and computational measures.The study uses linear finite elements, standard coarsening, full-weighting restriction, linear interpolation, and discretization coarse-grid approximation.Source: §3 · p. 14
  • For the second example, the study uses the exact solution 1 + sin(πx/2) sin(πy/2) exp(−π²t/2) and varies the time step from 0.04 to 0.001 at h = 2^-7.The boundary conditions are homogeneous Dirichlet conditions.Source: §3 · p. 14
  • The authors report satisfactory NTGW results in both examples, comparing averaged convergence factors, iterations to reduce the initial residual by 10^-10, CPU time, and error norms.For convergent schemes, the error norms are reported as constant across the three methods.Source: §3 · p. 14

4. Conclusions

The paper extends a new two-grid method to time-dependent problems using finite element spatial discretization. Its analysis uses the spectral radius, and the authors report a better averaged convergence factor than Gauss-Seidel and classical two-grid waveform relaxation.

4. Conclusions

  • The work extends the new two-grid method to time-dependent equations and uses finite elements, which the authors describe as yielding a more general ODE system than finite differences.The method is tested on the heat equation.Source: §4 · pp. 14–15
  • Across Tables 2 and 3, NTGW has a better averaged convergence factor than Gauss-Seidel and classical two-grid waveform relaxation.Table 2 compares the three methods at h = 2^-7 while τ varies from 0.04 to 0.001.Source: §4 · p. 15, Table 2 · p. 15
  • The convergence analysis is based on the spectral radius of the new two-grid waveform relaxation operator.The paper states that Lemmas 2.6 and 2.7 analyze NTGW's convergence property.Source: §4 · p. 15
  • The study addresses the continuous-time case; the authors identify the discontinuous-time case as future work and note that NTGW may be used for other time-dependent equations.Source: §4 · p. 15
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