# Problem on implementing fixed point method for nonlinear elliptic problem

**URL:** <https://community.freefem.org/t/problem-on-implementing-fixed-point-method-for-nonlinear-elliptic-problem/2322>\
**Category:** General Discussion\
**Created:** [March 9, 2023, 6:48am UTC](https://community.freefem.org/t/problem-on-implementing-fixed-point-method-for-nonlinear-elliptic-problem/2322 "2023-03-09T06:48:29Z")\
**Posts on this page:** 4\
**Page:** 1

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**Author:** ![AbuAzzan](https://yyz2.discourse-cdn.com/flex030/user_avatar/community.freefem.org/abuazzan/32/1645_2.png) [@AbuAzzan](https://community.freefem.org/u/AbuAzzan)\
**Post date:** [March 9, 2023, 6:48am UTC](https://community.freefem.org/t/problem-on-implementing-fixed-point-method-for-nonlinear-elliptic-problem/2322/1 "2023-03-09T06:48:29Z")

</div>

Hello freefem community,  
I am trying to solve a semilinear elliptic problem where the forcing term f(u) depends on the solution. Here I am using fixed point method implemented for nonlinear Schwarz method to solve the original problem. Unfortunately, my code is running but does not show any outputs. I guess there is an error in the fixed point method’s loop coding. Can someone comment on my problem and repair the error? With thanks.  
Here is my code attempt:

int n=0; //iteration  
int i=0; //number of meshes  
func g= x+y; //DIRICHELET boundary condition  
real x0=0, x1=9./16., y0=0, y1=1; //SUB-DOMAIN1 Dimensions  
real xx0=7./16., xx1=1, yy0=0, yy1=1; //SUB-DOMAIN2 Dimensions

// Size of both meshes  
real[int] h1(5) , h2(5); //size of both meshes  
real Globalnorm=1.;

//SUB-DOMAIN1 borders  
border AB1(t=x0, x1){x=t; y=y0; label=1;};  
border BC1(t=y0, y1){x=x1; y=t; label=2;};  
border CD1(t=x1, x0){x=t; y=y1; label=3;};  
border DA1(t=y1, y0){x=x0; y=t; label=4;};  
//SUB-DOMAIN2 borders  
border AB2(t=xx0, xx1){x=t; y=yy0; label=1;};  
border BC2(t=yy0, yy1){x=xx1; y=t; label=2;};  
border CD2(t=xx1, xx0){x=t; y=yy1; label=3;};  
border DA2(t=yy1, yy0){x=xx0; y=t; label=4;};

mesh Th1=square(2, 2, [x0+(x1-x0)\*x, y0+(y1-y0)\*y]); //initial Rectangular mesh of SUB-DOMAIN1  
mesh Th2=square(2, 2, [xx0+(xx1-xx0)\*x, yy0+(yy1-yy0)\*y]); //initial Rectangular mesh of SUB-DOMAIN2

//4 MESHES OF BOTH SUB-DOMAINS (Two choices of meshing)  
for (int i=0; i\<=4; i++){

h1[i]=(1./8.)_(1./(2.^i));  
h2[i]=(1./4.)_(1./(2.^i));  
Th1 = adaptmesh(Th1, h1[i], IsMetric=1);  
Th2 = adaptmesh(Th2, h2[i], IsMetric=1);

//Finite element spaces  
fespace Vh1(Th1, P1);  
Vh1 uh1, vh1, f1, R1, uold1=0, diff1;

fespace Vh2(Th2, P1);  
Vh2 uh2, vh2, f2, R2, uold2=0, diff2;

// plot to see the 2 meshes.  
plot(Th1, Th2, wait=true);

```
//solve both nonlinear subproblems

R1=-1/(1+0.25*uold1^2);
f1=uold1*V1;

```

problem Schw1(uh1, vh1, solver=CG, eps=1.e-6)  
=int2d(Th1)(dx(uh1)\*dx(vh1)+dy(uh1)_dy(vh1)) //bilinear term  
+int2d(Th1)(f1_vh1) //right hand side(nonlinear term )  
+on(1,3,4, uh1=g)+on(2, uh1=uold2); //external+internal boundaries

```
R2=-1/(1+0.25*uold2^2);
f2=uold2*V2;

```

problem Schw2(uh2, vh2, solver=CG, eps=1.e-6)  
=int2d(Th2)(dx(uh2)\*dx(vh2)+dy(uh2)_dy(vh2)) //bilinear term  
+int2d(Th2)(f2_vh2) //right hand side (nonlinear term)  
+on(1,2,3, uh2=g)+on(4, uh2=uh1); //external+internal boundarie

for (int n=1; n\<=35; n++){

```
while ( Globalnorm >= 1e -6 ) {	//fixed point loop for both subproblems 
Schw1;
Schw2;
V1 = -1/(1+0.25*uh1^2); // actualization	
V2 = -1/(1+0.25*uh2^2); // actualization
uold1=uh1; 
uold2=uh2;
};

```

diff1 = abs(uh1-uold1);  
norm1=diff1.max;  
diff2 = abs(uh2-uold2);  
norm2=diff2.max;  
Globalnorm=max(norm1, norm2);  
cout\<\<" --@ iteration no. --“\<\<n\<\<endl;  
cout\<\<“Difference error on subdomain 1 at iteration”\<\<”(“\<\<n\<\<”) =“\<\<norm1\<\<endl;  
cout\<\<“Difference error on subdomain 2 at iteration”\<\<”(“\<\<n\<\<”) ="\<\<norm2\<\<endl;  
};  
plot(uh1,uh2, wait=true);

};

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<div class="post-metadata">

**Author:** ![frederichecht](https://yyz2.discourse-cdn.com/flex030/user_avatar/community.freefem.org/frederichecht/32/15_2.png) [@frederichecht](https://community.freefem.org/u/frederichecht)\
**Post date:** [March 10, 2023, 4:11pm UTC](https://community.freefem.org/t/problem-on-implementing-fixed-point-method-for-nonlinear-elliptic-problem/2322/2 "2023-03-10T16:11:01Z")

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in you code what is V1 and V2 variable.

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<div class="post-metadata">

**Author:** ![AbuAzzan](https://yyz2.discourse-cdn.com/flex030/user_avatar/community.freefem.org/abuazzan/32/1645_2.png) [@AbuAzzan](https://community.freefem.org/u/AbuAzzan)\
**Post date:** [March 11, 2023, 10:16am UTC](https://community.freefem.org/t/problem-on-implementing-fixed-point-method-for-nonlinear-elliptic-problem/2322/3 "2023-03-11T10:16:00Z")

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Thank you for your reply Dr. Hecht

V1 and V2 are considered to be functions depend on uh1 and uh2 respectively

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<div class="post-metadata">

**Author:** ![AbuAzzan](https://yyz2.discourse-cdn.com/flex030/user_avatar/community.freefem.org/abuazzan/32/1645_2.png) [@AbuAzzan](https://community.freefem.org/u/AbuAzzan)\
**Post date:** [March 14, 2023, 5:41pm UTC](https://community.freefem.org/t/problem-on-implementing-fixed-point-method-for-nonlinear-elliptic-problem/2322/4 "2023-03-14T17:41:06Z")

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I am sorry for the unintentional typo mistake of V1 and V2. They are just written as R1 and R2 instead. However, the problem is still unsolved.

R1=-1/(1+0.25_uold1^2);  
f1=uold1_R1;  
////////////////////  
R2=-1/(1+0.25_uold2^2);  
f2=uold2_R2;  
///////////////////  
while ( Globalnorm \>= 1e -6 ) { //fixed point loop for both subproblems  
Schw1;  
Schw2;  
R1 = -1/(1+0.25_uh1^2); // actualization   
R2 = -1/(1+0.25_uh2^2); // actualization  
uold1=uh1;  
uold2=uh2;