From 7baa61dfb86765fd4ef2eb3a54c1753eb56bac25 Mon Sep 17 00:00:00 2001 From: Wolfgang Bangerth Date: Tue, 10 Feb 2015 17:27:48 -0600 Subject: [PATCH] Update a couple of formulas that gave Lei a fit (because, admittedly, they were not particularly clear). --- examples/step-33/doc/intro.dox | 16 ++++++++-------- 1 file changed, 8 insertions(+), 8 deletions(-) diff --git a/examples/step-33/doc/intro.dox b/examples/step-33/doc/intro.dox index c35476df7b..86c57e266b 100644 --- a/examples/step-33/doc/intro.dox +++ b/examples/step-33/doc/intro.dox @@ -164,19 +164,19 @@ $\frac{h}{2\delta T}$ with $h$ the diameter of the face to which the flux is applied, and $\delta T$ the current time step. With these choices, equating the residual to zero results in a -nonlinear system of equations which we solve the nonlinear system by a -Newton iteration, i.e. by iterating +nonlinear system of equations $R(\mathbf{W}_{n+1})=0$. We solve this nonlinear system by a +Newton iteration (in the same way as explained in step-15), i.e. by iterating @f{eqnarray*} -R'(\mathbf{W}^k,\delta \mathbf{W})(\mathbf z) & = & - -R(\mathbf{W}^{k})(\mathbf z) \qquad \qquad \forall \mathbf z\in V_h \\ -\mathbf{W}^{k+1} &=& \mathbf{W}^k + \delta \mathbf{W}, +R'(\mathbf{W}^k_{n+1},\delta \mathbf{W}_{n+1}^k)(\mathbf z) & = & - +R(\mathbf{W}^{k}_{n+1})(\mathbf z) \qquad \qquad \forall \mathbf z\in V_h \\ +\mathbf{W}^{k+1}_{n+1} &=& \mathbf{W}^k_{n+1} + \delta \mathbf{W}^k_{n+1}, @f} -until $|R(\mathbf{W}^k)|$ (the residual) is sufficiently small. By +until $|R(\mathbf{W}^k_{n+1})|$ (the residual) is sufficiently small. By testing with the nodal basis of a finite element space instead of all $\mathbf z$, we arrive at a linear system for $\delta \mathbf W$: @f{eqnarray*} -\mathbf R'(\mathbf{W}^k)\delta \mathbf{W} & = & - -\mathbf R(\mathbf{W}^{k}). +\mathbf R'(\mathbf{W}^k_{n+1})\delta \mathbf{W}^k_{n+1} & = & - +\mathbf R(\mathbf{W}^{k}_{n+1}). @f} This linear system is, in general, neither symmetric nor has any particular definiteness properties. We will either use a direct solver -- 2.39.5