# "NDSolve::femper" Error in Solving Coupled Nonlinear PDEs [duplicate]

Bug introduced in 12.0.0 - Fixed in Version: 12.2.0

For example, consider the following piar of PDEs with two unknown functions $$u(x,y)$$ and $$v(x,y)$$: $$\partial_x^2 u+\partial_y\left(v\partial_y u\right)=0\,,\quad\partial_x^2 v+\partial_y\left(u\partial_y v\right)=0\,,$$ as well as some suitable boundary conditions. We could try the following code

Eq1 = Inactive[Div][{{1, 0}, {0, v[x, y]}}.Inactive[Grad][u[x, y], {x, y}], {x, y}]==0;
Eq2 = Inactive[Div][{{1, 0}, {0, u[x, y]}}.Inactive[Grad][v[x, y], {x, y}], {x, y}]==0;
NDSolve[{Eq1, Eq2, DirichletCondition[u[x, y] == Cos[x + y], True], DirichletCondition[v[x, y] == Cos[x + y], True]}, {u[x, y], v[x, y]}, {x, y} \[Element] Disk[]]


Then we would have an error

NDSolve::femper: PDE parsing error of Div[{{1,0},{0,u}}.Grad\$146922]. Inconsistent equation dimensions.


In contrast, if we try to solve another similar pair of PDEs $$\partial_x^2 u+\partial_y\left(u\partial_y u\right)=0\,,\quad\partial_x^2 v+\partial_y\left(v\partial_y v\right)=0\,,$$ with similar codes, no error is reported and the codes run well.

More generally, as long as the coefficients of the second derivatives of one unknown function depend on other unknown functions, this error comes out.

What should I do to avoid this error?

• This is the same issue as in the other report. It's a bug and fixed in the upcoming version 12.2. I am sorry about the inconvenience this causes. Commented Nov 12, 2020 at 8:16
• To avoid this in versions before 12.2 you'd need to use InitializePDECoefficients and the low level fem code. Commented Nov 12, 2020 at 8:17