Optimized Ventcel-Schwarz waveform relaxation and mixed hybrid finite element method for transport problems

Thi-Thao-Phuong Hoang

doi:10.3934/dcdss.2022060

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2022, Volume 15, Issue 10: 2945-2964. Doi: 10.3934/dcdss.2022060

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Optimized Ventcel-Schwarz waveform relaxation and mixed hybrid finite element method for transport problems

Thi-Thao-Phuong Hoang

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

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

Received: June 30, 2021

Revised: December 31, 2021

Early access: March 2022

Published: September 2022

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

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Abstract

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.

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Mathematics Subject Classification: 65M55, 65M60, 65M50.

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Figure 1. A decomposition of $ \Omega $ into two nonoverlapping subdomains

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Figure 2. Nonconforming time grids in the subdomains

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Figure 3. [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

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Figure 4. [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

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Figure 5. [Test case 2] Errors in the concentration $ c $ (left) and the vector field $ \pmb{\varphi} $ (right) between the reference and multidomain solutions

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Table 1. [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]
$L^{2}$ errors	$\mathit{\boldsymbol{\varphi }}$	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

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Table 2. [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]
$ L^{2} $ errors	$ \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

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Table 3. [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 $

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