I am solving a nonlinear Poisson's equation numerically using a mixed Chebyshev/Fourier spectral methods. Thus, assuming $x$ is periodic and $y$ is nonperiodic. I am trying to test my current numerical method/solver by using the method of manufacturing solution (MMS). Additionally, I am applying some coordinate mapping or rescaling my Chebyshev Differentiation Matrices to an arbitrary domain [a,b] by performing a change of variables as the following: $$ y_L = (1/2)\left[ (b-a)y+(b+a) \right] $$ Thus, my chebyshev derivatives can be rewritten as: $$ D_L = D \left[ \frac{2}{b-a} \right] $$ The two above expressions were taken from the Weideman and Reddy paper found in this post: Are there any alterations for the Chebyshev Differentiation Matrices on an arbitrary domain [a,b]?
I) As an initial step I solved the linear Poisson's equation with the rescaled Chebyshev derivatives: $$ \nabla^2 u = S $$ where BCs are periodic in x and zero Dirichlet in y as $u(x,a)=0$ & $u(x,b)=0$.
For the simple above case the test I performed with the MMS is: $$ u(x,y)=(y-a)(y-b)\sin{Ax} $$ The above case works perfectly for me and I have no issue at all.
II) However, when I try to do the same for the nonlinear Poisson's equation with following expression: $$ \nabla \cdot (n\nabla u) = S $$ or can be expanded as: $n \nabla^2 u + \nabla u \cdot \nabla n = S$
Then, using the MMS I chose $u(x,y)$ and $n(x,y)$ similarly as the first case: $$ u(x,y)=(y-a)(y-b)\sin{Ax} $$ $$ n(x,y)=y^2\sin{3Ax}+10 $$ The above expressions were chosen sort of in an arbitrary way so that they satisfy my BCs. However, the above case with the nonlinearity added (i.e. $n(x,y)$) does not work for me and I am trying to figure out if my choice of $n(x,y)$ is incorrect or if something else is going on in my numerical solver that I can fix. Is there a more robust way to test the second case? Thanks.
