Study guide: Solving nonlinear ODE and PDE problems

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Algorithm for relaxed Picard iteration

Given $ A(u)u=b(u) $ and an initial guess $ u^{-} $, iterate until convergence:

solve $ A(u^{-})u^* = b(u^{-}) $ with respect to $ u^* $

$ u = \omega u^* + (1-\omega) u^{-} $

$ u^{-}\ \leftarrow\ u $

"Until convergence": $ ||u - u^{-}|| \leq \epsilon_u $ or $ ||A(u)u-b|| \leq\epsilon_r $