American Institute of Mathematical Sciences

doi: 10.3934/dcdss.2022060
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Optimized Ventcel-Schwarz waveform relaxation and mixed hybrid finite element method for transport problems

 Department of Mathematics and Statistics, Auburn University, Auburn, AL 36849, USA

Dedicated to Professor Georg Hetzer on the occasion of his 75th birthday

Received  July 2021 Revised  January 2022 Early access March 2022

Fund Project: This work is partially supported by the US National Science Foundation under grant number DMS-1912626

This paper is concerned with the optimized Schwarz waveform relaxation method and Ventcel transmission conditions for the linear advection-diffusion equation. A mixed formulation is considered in which the flux variable represents both diffusive and advective flux, and Lagrange multipliers are introduced on the interfaces between nonoverlapping subdomains to handle tangential derivatives in the Ventcel conditions. A space-time interface problem is formulated and is solved iteratively. Each iteration involves the solution of time-dependent problems with Ventcel boundary conditions in the subdomains. The subdomain problems are discretized in space by a mixed hybrid finite element method based on the lowest-order Raviart-Thomas space and in time by the backward Euler method. The proposed algorithm is fully implicit and enables different time steps in the subdomains. Numerical results with discontinuous coefficients and various Peclét numbers validate the accuracy of the method with nonconforming time grids and confirm the improved convergence properties of Ventcel conditions over Robin conditions.

Citation: Thi-Thao-Phuong Hoang. Optimized Ventcel-Schwarz waveform relaxation and mixed hybrid finite element method for transport problems. Discrete and Continuous Dynamical Systems - S, doi: 10.3934/dcdss.2022060
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References:
A decomposition of $\Omega$ into two nonoverlapping subdomains
Nonconforming time grids in the subdomains
[Test case 2] Convergence curves by Jacobi (left) and GMRES (right) for different Péclet numbers: $L^{2}-$norm errors in the concentration at $T = 1$ with optimized two-sided Robin (blue curves), optimized one-sided Ventcel (magenta curves), and optimized weighted Ventcel (red curves) parameters
[Test case 2: Advection dominance] Level curves for the error in concentration (in logarithmic scales) after 12 iterations of Jacobi (left) and GMRES (right) for various values of $p$ and $q$. The red star shows the optimized values computed by numerically minimizing the continuous convergence factor of the OSWR algorithm
[Test case 2] Errors in the concentration $c$ (left) and the vector field $\pmb{\varphi}$ (right) between the reference and multidomain solutions
[Test case 1] Accuracy in space (the convergence rates are shown in square brackets) and numbers of Jacobi and GMRES iterations for different optimized parameters
 $h$ $1/20$ $1/40$ $1/80$ $1/160$ $L^{2}$ errors $c$ 0.0641 0.0321 [1.00] 0.0160 [1.00] 0.0080 [1.00] $\mathit{\boldsymbol{\varphi }}$ 0.0453 0.0227 [1.00] 0.0114 [0.99] 0.0057 [1.00] Jacobi 2-sided Robin 21 21 23 25 1-sided Ventcel 11 11 12 13 weighted Ventcel 11 11 12 13 GMRES 2-sided Robin 16 16 20 22 1-sided Ventcel 10 11 11 12 weighted Ventcel 8 10 10 11
 $h$ $1/20$ $1/40$ $1/80$ $1/160$ $L^{2}$ errors $c$ 0.0641 0.0321 [1.00] 0.0160 [1.00] 0.0080 [1.00] $\mathit{\boldsymbol{\varphi }}$ 0.0453 0.0227 [1.00] 0.0114 [0.99] 0.0057 [1.00] Jacobi 2-sided Robin 21 21 23 25 1-sided Ventcel 11 11 12 13 weighted Ventcel 11 11 12 13 GMRES 2-sided Robin 16 16 20 22 1-sided Ventcel 10 11 11 12 weighted Ventcel 8 10 10 11
[Test case 1] Accuracy in time (the convergence rates are shown in square brackets) and numbers of Jacobi and GMRES iterations for different optimized parameters
 $\Delta t_{2}$ $T/6$ $T/12$ $T/24$ $T/48$ $L^{2}$ errors $c$ 0.1859 0.0708 [1.39] 0.0301 [1.23] 0.0145 [1.05] $\pmb{\varphi}$ 0.2008 0.0768 [1.39] 0.0325 [1.24] 0.0150 [1.12] Jacobi 2-sided Robin 33 33 33 35 1-sided Ventcel 17 17 17 17 weighted Ventcel 17 17 17 17 GMRES 2-sided Robin 18 18 20 24 1-sided Ventcel 11 12 13 14 weighted Ventcel 10 11 12 13
 $\Delta t_{2}$ $T/6$ $T/12$ $T/24$ $T/48$ $L^{2}$ errors $c$ 0.1859 0.0708 [1.39] 0.0301 [1.23] 0.0145 [1.05] $\pmb{\varphi}$ 0.2008 0.0768 [1.39] 0.0325 [1.24] 0.0150 [1.12] Jacobi 2-sided Robin 33 33 33 35 1-sided Ventcel 17 17 17 17 weighted Ventcel 17 17 17 17 GMRES 2-sided Robin 18 18 20 24 1-sided Ventcel 11 12 13 14 weighted Ventcel 10 11 12 13
[Test case 2] Discontinuous diffusion and advection coefficients
 Problems $d_{1}$ $\pmb{u}_{1}$ ${\rm{Pe}}_{G,1}$ $d_{2}$ $\pmb{u}_{2}$ ${\rm{Pe}}_{G,2}$ (a) Diffusion dominance $1$ $(-0.02, \; -0.5)^{T}$ $\approx 0.5$ $0.1$ $(-0.02, \; -0.05)^{T}$ $\approx 0.5$ (b) Mixed regime $0.01$ $(-0.02, \; -0.5)^{T}$ $\approx 50$ $0.1$ $(-0.02, \; -0.05)^{T}$ $\approx 0.5$ (c) Advection dominance $0.02$ $(0.5, \; 1)^{T}$ $\approx 56$ $0.002$ $(0.5, \; 0.1)^{T}$ $\approx 255$
 Problems $d_{1}$ $\pmb{u}_{1}$ ${\rm{Pe}}_{G,1}$ $d_{2}$ $\pmb{u}_{2}$ ${\rm{Pe}}_{G,2}$ (a) Diffusion dominance $1$ $(-0.02, \; -0.5)^{T}$ $\approx 0.5$ $0.1$ $(-0.02, \; -0.05)^{T}$ $\approx 0.5$ (b) Mixed regime $0.01$ $(-0.02, \; -0.5)^{T}$ $\approx 50$ $0.1$ $(-0.02, \; -0.05)^{T}$ $\approx 0.5$ (c) Advection dominance $0.02$ $(0.5, \; 1)^{T}$ $\approx 56$ $0.002$ $(0.5, \; 0.1)^{T}$ $\approx 255$
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