I need to solve the following system of 2 coupled non-linear PDEs, desirably on a domain of arbitrary shape, but it would be a nice start if it was possible on a simple rectangular domain with Dirichlet boundary conditions imposed for $u(x,y)$, $w(x,y)$. The system is
eq1 = d^2 a1^2 (D[(D[w[x,y],x]/q),x] + D[(D[w[x, y],y]/q),y]) + q/d^2 (w[x,y]-u[x, y])+
a1((a0+g0) + a1 w[x,y] + g1 u[x,y]) - q k1 w[x,y]==0
eq2 = D[u[x,y],x,x] + D[u[x,y],y,y]+q/d^2 (w[x,y]-u[x,y]) + m1 q +
g1 ((a0+g0) + a1 w[x,y] + g1 u[x,y])==0
and the nonlinearity comes from the form of $q$ that is
q = (d^2 k0 + d^2 m0 + u[x,y] + d^2 m1 u[x,y] - w[x,y] + d^2 k1 w[x,y])/(d^2 q0)
d,a0,g0,a1,g1,m0,m1,k0,k1,q0
are arbitrary constants.
I tried the following but it doesn't work
Needs["NDSolve`FEM`"];
Q = ToElementMesh[Rectangle[{0,0}, {1, 1}], "MaxBoundaryCellMeasure" -> 0.005,"MeshElementType" -> TriangleElement];
d = 0.1; k0 = 0; a0 = 0; g0 = 0; g1 = 0.5; a1 = 0.5; q0 = 10; k1 =0.3; m0 = 1; m1 = 0.5;
q = (d^2 k0 + d^2 m0 + u[x,y] + d^2 m1 u[x,y] - w[x,y] + d^2 k1 w[x,y])/(d^2 q0);
eq1 = d^2 a1^2 ( D[(D[w[x, y], x]/q), x] + D[(D[w[x, y], y]/q), y]) + q/d^2 (w[x, y] - u[x, y]) + a1 ((a0 + g0) + a1 w[x, y] + g1 u[x, y]) - q k1 w[x, y];
eq2 = D[u[x, y], x, x] + D[u[x, y], y, y] + q/d^2 (w[x, y] - u[x, y]) + m1 q + g1 ((a0 + g0) + a1 w[x, y] + g1 u[x, y]);
sol = NDSolveValue[{{eq1 == 0, eq2 == 0}, DirichletCondition[{u[x, y] == 0,w[x, y] == 1}, True]}, {u,w}, {x,y} \[Element] Q]
MethodOfLines
. This may help, if you can reformulate the problem such that your desired solution is equal to a fixed point of soome dynamic problem. I cannot tell aproiory, if it is the case though. $\endgroup$InitialSeeding"->{u[x,y]==uGuess,w[x,y]==wGuess}
. $\endgroup$