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In brief, I have a non-linear equation of the form $$\nabla \big(n(F) \nabla F\big) = 0$$ defined on a 2D domain, and I'm trying to solve it using simple Galerkin FEM with Lagrange P1 elements on a triangular mesh with Newton-Raphson method. I'm facing a problem when I solve an equation $\hat{J} \delta F = R$ for the step $\delta F$.

Suppose the 2D domain is the square $[0,1]\times [0,1]$ and the boundary conditions are: $F(x) = F_1$ on the left edge, $F(x) = F_2$ on the right edge ($F_1$ and $F_2$ are just constants), and $\partial_n F(x) = 0$ on the top and the bottom edges. The function $n(F)$ is smooth.

Linearizing the above equation at $F=F_0$ (my initial guess), the following equation for the step $\delta F$ can be obtained: $$\nabla(n_0'\ \delta F\ \nabla F_0 + n_0 \nabla \delta F) = -\nabla(n_0 \nabla F_0), \quad n_0 = n(F_0).$$

When the difference $\lvert F_1 - F_2 \rvert$ is small, the method works fine. Otherwise, the solution $\delta F$ of this equation has spurious oscillations which resemble those when Galerkin FEM is applied to convection dominated problems (and this equation is of that type when $\lvert F_1 - F_2 \rvert$ and hence $\lvert \nabla F_0 \rvert$ are not small).

My question is: should I apply some stabilization scheme when I try to use Newton's method? I've checked several books on non-linear FEM and none of them mention such problems. If yes, is it possible to avoid the necessity of stabilization?

I would appreciate any references to the literature where such issues are discussed.

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