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$s_k$ is the "approximate Newton" search direction. So in essence, when they say Choose $s_k$ such that $\|F(x_k)+F'(x_k)s_k\| \le \eta_k \|F(x_k)\|$ they are saying: Solve the Newton system $F'(x_k)s_k = - F(x_k)$ inexactly for $s_k$ until the norm of the residual $F(x_k)+F'(x_k)s_k$ is smaller than the norm of the right hand side $-F(x_k)$ by a ...


You can arrive at the Jacobian analytically it just takes a few steps So assuming we have our typical FE field values: $$ u_i = \sum_j \phi^j u_i^j $$ Where $i$ represents coordinate direction, $\phi$ our shape function, and $j$ index to DOF. Start with our stabilization parameter, assuming you are using Euclidian norm: $$ \tau = \sqrt{\alpha \lVert u \...


Yes, of course you can. You end up with something that is often called "mixed finite elements" if, for example, you are considering a formulation that involves velocities and pressures.

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