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A second-order correction method for loosely coupled discretizations applied to parabolic-parabolic interface problems

Erik Burman, Rebecca Durst, Miguel A. Fernández, Johnny Guzmán, Sijing Liu

Abstract

We consider a parabolic-parabolic interface problem and construct a loosely coupled prediction-correction scheme based on the Robin-Robin splitting method analyzed in [J. Numer. Math., 31(1):59--77, 2023]. We show that the errors of the correction step converge at $\mathcal O((Δt)^2)$, under suitable convergence rate assumptions on the discrete time derivative of the prediction step, where $Δt$ stands for the time-step length. Numerical results are shown to support our analysis and the assumptions.

A second-order correction method for loosely coupled discretizations applied to parabolic-parabolic interface problems

Abstract

We consider a parabolic-parabolic interface problem and construct a loosely coupled prediction-correction scheme based on the Robin-Robin splitting method analyzed in [J. Numer. Math., 31(1):59--77, 2023]. We show that the errors of the correction step converge at , under suitable convergence rate assumptions on the discrete time derivative of the prediction step, where stands for the time-step length. Numerical results are shown to support our analysis and the assumptions.
Paper Structure (21 sections, 9 theorems, 82 equations, 3 figures, 4 tables)

This paper contains 21 sections, 9 theorems, 82 equations, 3 figures, 4 tables.

Table of Contents

  1. Introduction
  2. Prediction-correction method for a parabolic\\ parabolic interface problem
  3. Variational form
  4. Defect-correction method
  5. Prediction step
  6. Correction step
  7. Estimates for the prediction step
  8. Error analysis for the correction step
  9. Error equations
  10. Preliminary estimate for the correction step
  11. Estimate for $F_1^{n+1}$
  12. Estimate for $R^{n+1}$
  13. Estimate for $T^{n+1}$
  14. Estimate for $\frac{\Delta t}{\alpha}\bigl\langle \Lambda_1^{n+1}, \Lambda_0^{n+1}-\Lambda_0^{n} \bigr\rangle$
  15. Main result: error estimate
  16. ...and 6 more sections

Key Result

Corollary 3.2

We have, for any $n=0,1,2,\ldots, N-1$.

Figures (3)

  • Figure 1: The domains $\Omega_f$ and $\Omega_s$ with interface $\Sigma$ and Neumann boundaries.
  • Figure 2: The domains $\Omega_f$ and $\Omega_s$ with horizontal interface $\Sigma$
  • Figure 3: The domains $\Omega_f$ and $\Omega_s$ with non-horizontal interface $\Sigma$.

Theorems & Definitions (23)

  • Corollary 3.2
  • proof
  • Remark 3.3
  • Lemma 4.1
  • proof
  • Proposition 4.2
  • Lemma 4.3
  • proof
  • Lemma 4.4
  • proof
  • ...and 13 more