The application is a simple non-linear advection diffusion problem (steady state) using DGFEM. My error at each iteration is given by $$ e_{n+1} = ||\mathbf{J}^{-1}(\mathbf{u}_{n})\mathbf{F}(\mathbf{u}_{n})||_{L^{2}(\Omega)} = ||\mathbf{u}_{n+1} - \mathbf{u}_{n}||_{L^{2}(\Omega)} $$
For testing purposes, I'm using an $L^{2}$ projection of the exact solution as an initial guess, i.e. solve: $$ \sum_{K\in\mathcal{M}_{h}}\int_{K}uv = \sum_{K\in\mathcal{M}_{h}}\int_{K}u_{\text{exact}}v \implies \mathbf{M}\mathbf{u}_{0} = \mathbf{f} $$
I see perfect quadratic convergence for the steady state nonlinear diffusion problem, and for most cases of nonlinear advection diffusion. For some cases it takes a VERY large number of iterations before I see quadratic convergence. It's perplexing since the initial guess is essentially the solution I want to end up at...
1) Does the advective term mess with the convergence rate of Newton's Method? Or is this just a sign that I've computed the Jacobian part of the advective terms incorrectly? I know the quadratic rate is only locally, but seeing as how the initial guess IS the solution...
2) If I start with an initial guess of $\mathbf{u}_{0}$ using the $L^{2}$ projection as defined above, should the method converge immediately or only in the case of the exact solution being exactly represented by the DG basis? Some numerical experiments point to the latter.
Here is one simple example. The model problem is $$ \nabla\cdot(\mathbf{s}(u)u) - \nabla\cdot(\kappa(u)\nabla u) = f \;\;\;\text{in}\;\Omega\subseteq\mathbb{R}^{2}\\ u = g_{D} \;\;\;\text{on}\;\partial\Omega $$
I take the exact solution $u(x,y) = x(x-1)y(y-1)$, $\mathbf{s} = [1+u,1-u]$ and $\kappa(u) = 1+u$ on the domain $\Omega = [-1,1]^{2}$. The initial guess is taken to be $\mathbf{u}_{0}$ from the $L^{2}$ projection as described above.
For basis degree p=1, here is the result:
Refinement Level 0
Iteration: 1, Error: 4.001769e+00
Iteration: 2, Error: 1.828048e+00
Iteration: 3, Error: 6.720810e+00
Iteration: 4, Error: 5.483628e+00
Iteration: 5, Error: 6.469472e+00
Iteration: 6, Error: 2.341527e+00
Iteration: 7, Error: 4.989268e+00
Iteration: 8, Error: 4.963194e+01
Iteration: 9, Error: 2.479127e+01
Iteration: 10, Error: 1.239452e+01
Iteration: 11, Error: 6.190117e+00
Iteration: 12, Error: 3.034077e+00
Iteration: 13, Error: 1.342876e+00
Iteration: 14, Error: 8.747657e-01
Iteration: 15, Error: 5.244396e-01
Iteration: 16, Error: 1.740700e-01
Iteration: 17, Error: 5.298694e-03
Iteration: 18, Error: 1.344089e-05
Iteration: 19, Error: 2.878889e-11
Iteration: 20, Error: 8.722883e-16
Iterations for convergence: 20
Refinement Level 1
Iteration: 1, Error: 1.241453e+00
Iteration: 2, Error: 4.959870e-01
Iteration: 3, Error: 1.097905e-01
Iteration: 4, Error: 6.880627e-03
Iteration: 5, Error: 5.445979e-05
Iteration: 6, Error: 3.168481e-09
Iteration: 7, Error: 1.874668e-15
Iterations for convergence: 7
Refinement Level 2
Iteration: 1, Error: 2.627395e-01
Iteration: 2, Error: 2.936349e-02
Iteration: 3, Error: 6.427106e-04
Iteration: 4, Error: 3.024645e-07
Iteration: 5, Error: 2.557842e-13
Iterations for convergence: 5
Refinement Level 3
Iteration: 1, Error: 5.830850e-02
Iteration: 2, Error: 1.113766e-03
Iteration: 3, Error: 8.567317e-07
Iteration: 4, Error: 5.108461e-13
Iterations for convergence: 4
Refinement Level 4
Iteration: 1, Error: 1.372896e-02
Iteration: 2, Error: 6.526761e-05
Iteration: 3, Error: 2.784184e-09
Iteration: 4, Error: 1.217915e-16
Iterations for convergence: 4
Refinement Level 5
Iteration: 1, Error: 3.357581e-03
Iteration: 2, Error: 4.495462e-06
Iteration: 3, Error: 1.148118e-11
Iteration: 4, Error: 1.162268e-16
Iterations for convergence: 4
Convergence in L2-norm:
1.511590
1.809664
2.135559
2.080259
2.030937
Convergence in H1-norm:
0.748175
1.164069
1.155149
1.067994
1.029409
Then for $p=2$, Newton's method on the coarsest mesh takes an absurd number of iterations to converge...
Refinement Level 0
Iteration: 1, Error: 9.021838e-01
Iteration: 2, Error: 6.832202e-01
Iteration: 3, Error: 2.971411e-01
...
Iteration: 683, Error: 1.499679e+00
Iteration: 684, Error: 8.019582e-01
Iteration: 685, Error: 4.119762e-01
Iteration: 686, Error: 1.051567e-01
Iteration: 687, Error: 5.846994e-03
Iteration: 688, Error: 1.780973e-05
Iteration: 689, Error: 4.898476e-11
Iteration: 690, Error: 1.755067e-15
Iterations for convergence: 690
Refinement Level 1
Iteration: 1, Error: 8.871096e-02
Iteration: 2, Error: 8.290874e-03
Iteration: 3, Error: 8.002375e-05
Iteration: 4, Error: 4.729235e-09
Iteration: 5, Error: 4.802363e-16
Iterations for convergence: 5
Refinement Level 2
Iteration: 1, Error: 1.225342e-02
Iteration: 2, Error: 9.032665e-05
Iteration: 3, Error: 1.068461e-08
Iteration: 4, Error: 1.175537e-15
Iterations for convergence: 4
Refinement Level 3
Iteration: 1, Error: 3.305251e-03
Iteration: 2, Error: 3.496646e-06
Iteration: 3, Error: 5.546949e-12
Iterations for convergence: 3
Refinement Level 4
Iteration: 1, Error: 9.875502e-04
Iteration: 2, Error: 3.438791e-07
Iteration: 3, Error: 5.191102e-14
Iterations for convergence: 3
Refinement Level 5
Iteration: 1, Error: 2.724040e-04
Iteration: 2, Error: 2.672095e-08
Iteration: 3, Error: 3.691891e-16
Iterations for convergence: 3
Convergence in L2-norm:
4.403329
2.867100
2.012923
1.775206
1.863933
Convergence in H1-norm:
3.108488
2.144271
2.073729
2.035578
2.017033
And for $p=4$, convergence is in a single iteration for the given initial guess. The error is just about zero as expected.
Refinement Level 0
Iteration: 1, Error: 5.726012e-12
Iterations for convergence: 1
Refinement Level 1
Iteration: 1, Error: 1.410337e-11
Iterations for convergence: 1
Refinement Level 2
Iteration: 1, Error: 2.317826e-12
Iterations for convergence: 1
Refinement Level 3
Iteration: 1, Error: 5.361553e-12
Iterations for convergence: 1
Refinement Level 4
Iteration: 1, Error: 4.668317e-13
Iterations for convergence: 1
Refinement Level 5
Iteration: 1, Error: 2.536992e-13
Iterations for convergence: 1