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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]
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  • $\begingroup$ For numerical solution, you need to provide data for the parameters/constants and also initial and boundary conditions. $\endgroup$
    – zhk
    Feb 4, 2017 at 13:24
  • $\begingroup$ Edit the question and include your try in MMA. $\endgroup$
    – zhk
    Feb 4, 2017 at 13:25
  • $\begingroup$ In addition to what MMM has already written, NDSolve does not support nonlinear equations, and yours is heavy nonlinear. One exclusion is the case of time-dependent equations that can be treated with the 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. $\endgroup$ Feb 4, 2017 at 13:50
  • $\begingroup$ In version 12 there is a nonlinear FEM solver; however, this equation does not converge. Do you have an initial guess / at starting vector to help find a solution? You could then specify that with InitialSeeding"->{u[x,y]==uGuess,w[x,y]==wGuess}. $\endgroup$
    – user21
    Apr 17, 2019 at 12:26

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