The steps above can be abbreviated as a fixed-point Gauss-Seidel problem
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| body | --uriencoded--\widetilde%7B\mathbf d%7D%5e%7Bn+1%7D_%7B\Gamma, i+1%7D = \mathbf H(\mathbf d%5e%7Bn+1%7D_%7B\Gamma,i%7D) |
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with respect to the interface displacement, which is to be solved until convergence of the corresponding residuals | Mathinline |
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| body | --uriencoded--\mathbf r_%7B\Gamma, i+1%7D = \mathbf R( \mathbf%7B d%7D%5e%7Bn+1%7D_%7B\Gamma, i%7D) = \mathbf H(\mathbf d%5e%7Bn+1%7D_%7B\Gamma, i+1%7D) - \mathbf%7B d%7D%5e%7Bn+1%7D_%7B\Gamma, i%7D= \widetilde%7B\mathbf d%7D%5e%7Bn+1%7D_%7B\Gamma, i+1%7D - \mathbf%7B d%7D%5e%7Bn+1%7D_%7B\Gamma, i%7D |
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. Typical convergence norms are | Mathinline |
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| body | --uriencoded--%7C%7C\mathbf r_%7B\Gamma, i+1%7D%7C%7C_2/ \sqrt%7BN_\Gamma%7D < \epsilon |
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which makes the norm independent on the interface mesh size or relative convergence | Mathinline |
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| body | --uriencoded--%7C%7C\mathbf r_%7B\Gamma,i+1%7D%7C%7C_2 / %7C%7C\widetilde%7B\mathbf d%7D_%7BG,i+1%7D %7C%7C_2 < \epsilon |
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. Note that the cost of evaluation of is high and it may contain noise/errors (for example due to not fully converged solvers).Various methods can be used to ensure and accelerated convergence of the root-finding problem. A (dynamic) under-relaxation is
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| body | --uriencoded--\mathbf%7B d%7D%5e%7Bn+1%7D_%7B\Gamma, i+1%7D = \mathbf%7B d%7D%5e%7Bn+1%7D_%7B\Gamma, i%7D + w_i \, \mathbf r_%7B\Gamma, i+1%7D = w_i \widetilde%7B\mathbf d%7D%5e%7Bn+1%7D_%7B\Gamma, i+1%7D + [1-w_i] \mathbf%7B d%7D%5e%7Bn+1%7D_%7B\Gamma, i%7D |
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. For example Aitken under-relaxation is given by | Mathinline |
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| body | --uriencoded--w_%7Bi+1%7D=-w_i \mathbf r_%7B\Gamma,i+1%7D \cdot \left[\mathbf r_%7B\Gamma,i+2%7D-\mathbf r_%7B\Gamma,i+1%7D\right]/%7C%7C\mathbf r_%7B\Gamma,i+2%7D-\mathbf r_%7B\Gamma,i+1%7D%7C%7C%5e2_2 |
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. | Mathblock |
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--uriencoded--\mathbf r_%7B\Gamma, i+1%7D = \mathbf R( \mathbf%7B d%7D%5e%7Bn+1%7D_%7B\Gamma, i%7D) =
\mathbf H(\mathbf d%5e%7Bn+1%7D_%7B\Gamma, i+1%7D) - \mathbf%7B d%7D%5e%7Bn+1%7D_%7B\Gamma, i%7D= \widetilde%7B\mathbf d%7D%5e%7Bn+1%7D_%7B\Gamma, i+1%7D - \mathbf%7B d%7D%5e%7Bn+1%7D_%7B\Gamma, i%7D |
