Domain decomposition
We present three classic examples of domain decomposition technique: first, Schwarz algorithm with overlapping, second Schwarz algorithm without overlapping (also call Shur complement), and last we show to use the conjugate gradient to solve the boundary problem of the Shur complement.
Schwarz overlapping
To solve:
the Schwarz algorithm runs like this:
where \(\Gamma_i\) is the boundary of \(\Omega_i\) and on the condition that \(\Omega_1\cap\Omega_2\neq\emptyset\) and that \(u_i\) are zero at iteration 1.
Here we take \(\Omega_1\) to be a quadrangle, \(\Omega_2\) a disk and we apply the algorithm starting from zero.

Tip
Schwarz overlapping
1 // Parameters
2 int inside =2; //inside boundary
3 int outside = 1; //outside boundary
4 int n = 4;
5
6 // Mesh
7 border a(t=1, 2){x=t; y=0; label=outside;}
8 border b(t=0, 1){x=2; y=t; label=outside;}
9 border c(t=2, 0){x=t; y=1; label=outside;}
10 border d(t=1, 0){x=1-t; y=t; label=inside;}
11 border e(t=0, pi/2){x=cos(t); y=sin(t); label=inside;}
12 border e1(t=pi/2, 2*pi){x=cos(t); y=sin(t); label=outside;}
13 mesh th = buildmesh(a(5*n) + b(5*n) + c(10*n) + d(5*n));
14 mesh TH = buildmesh(e(5*n) + e1(25*n));
15 plot(th, TH, wait=true); //to see the 2 meshes
16
17 // Fespace
18 fespace vh(th, P1);
19 vh u=0, v;
20
21 fespace VH(TH, P1);
22 VH U, V;
23
24 // Problem
25 int i = 0;
26 problem PB (U, V, init=i, solver=Cholesky)
27 = int2d(TH)(
28 dx(U)*dx(V)
29 + dy(U)*dy(V)
30 )
31 + int2d(TH)(
32 - V
33 )
34 + on(inside, U=u)
35 + on(outside, U=0)
36 ;
37
38 problem pb (u, v, init=i, solver=Cholesky)
39 = int2d(th)(
40 dx(u)*dx(v)
41 + dy(u)*dy(v)
42 )
43 + int2d(th)(
44 - v
45 )
46 + on(inside, u=U)
47 + on(outside, u=0)
48 ;
49
50 // Calculation loop
51 for (i = 0 ; i < 10; i++){
52 // Solve
53 PB;
54 pb;
55
56 // Plot
57 plot(U, u, wait=true);
58 }
Schwarz non overlapping Scheme
To solve:
the Schwarz algorithm for domain decomposition without overlapping runs like this

Let introduce \(\Gamma_i\) is common the boundary of \(\Omega_1\) and \(\Omega_2\) and \(\Gamma_e^i= \partial \Omega_i \setminus \Gamma_i\).
The problem find \(\lambda\) such that \((u_1|_{\Gamma_i}=u_2|_{\Gamma_i})\) where \(u_i\) is solution of the following Laplace problem:
To solve this problem we just make a loop with upgrading \(\lambda\) with
where the sign \(+\) or \(-\) of \({\pm}\) is choose to have convergence.
Tip
Schwarz non-overlapping
1 // Parameters
2 int inside = 2; int outside = 1; int n = 4;
3
4 // Mesh
5 border a(t=1, 2){x=t; y=0; label=outside;};
6 border b(t=0, 1){x=2; y=t; label=outside;};
7 border c(t=2, 0){x=t; y=1; label=outside;};
8 border d(t=1, 0){x=1-t; y=t; label=inside;};
9 border e(t=0, 1){x=1-t; y=t; label=inside;};
10 border e1(t=pi/2, 2*pi){x=cos(t); y=sin(t); label=outside;};
11 mesh th = buildmesh(a(5*n) + b(5*n) + c(10*n) + d(5*n));
12 mesh TH = buildmesh(e(5*n) + e1(25*n));
13 plot(th, TH, wait=true);
14
15 // Fespace
16 fespace vh(th, P1);
17 vh u=0, v;
18 vh lambda=0;
19
20 fespace VH(TH, P1);
21 VH U, V;
22
23 // Problem
24 int i = 0;
25 problem PB (U, V, init=i, solver=Cholesky)
26 = int2d(TH)(
27 dx(U)*dx(V)
28 + dy(U)*dy(V)
29 )
30 + int2d(TH)(
31 - V
32 )
33 + int1d(TH, inside)(
34 lambda*V
35 )
36 + on(outside, U= 0 )
37 ;
38
39 problem pb (u, v, init=i, solver=Cholesky)
40 = int2d(th)(
41 dx(u)*dx(v)
42 + dy(u)*dy(v)
43 )
44 + int2d(th)(
45 - v
46 )
47 + int1d(th, inside)(
48 - lambda*v
49 )
50 + on(outside, u=0)
51 ;
52
53 for (i = 0; i < 10; i++){
54 // Solve
55 PB;
56 pb;
57 lambda = lambda - (u-U)/2;
58
59 // Plot
60 plot(U,u,wait=true);
61 }
62
63 // Plot
64 plot(U, u);
Schwarz conjuguate gradient
To solve \(-\Delta u =f \;\mbox{in}\;\Omega=\Omega_1\cup\Omega_2\quad u|_\Gamma=0\) the Schwarz algorithm for domain decomposition without overlapping runs like this
Let introduce \(\Gamma_i\) is common the boundary of \(\Omega_1\) and \(\Omega_2\) and \(\Gamma_e^i= \partial \Omega_i \setminus \Gamma_i\).
The problem find \(\lambda\) such that \((u_1|_{\Gamma_i}=u_2|_{\Gamma_i})\) where \(u_i\) is solution of the following Laplace problem:
The version of this example uses the Shur complement. The problem on the border is solved by a conjugate gradient method.
Tip
Schwarz conjugate gradient
First, we construct the two domains:
1 // Parameters
2 int inside = 2; int outside = 1; int n = 4;
3
4 // Mesh
5 border Gamma1(t=1, 2){x=t; y=0; label=outside;}
6 border Gamma2(t=0, 1){x=2; y=t; label=outside;}
7 border Gamma3(t=2, 0){x=t; y=1; label=outside;}
8 border GammaInside(t=1, 0){x=1-t; y=t; label=inside;}
9 border GammaArc(t=pi/2, 2*pi){x=cos(t); y=sin(t); label=outside;}
10 mesh Th1 = buildmesh(Gamma1(5*n) + Gamma2(5*n) + GammaInside(5*n) + Gamma3(5*n));
11 mesh Th2 = buildmesh(GammaInside(-5*n) + GammaArc(25*n));
12 plot(Th1, Th2);
Now, define the finite element spaces:
1 // Fespace
2 fespace Vh1(Th1, P1);
3 Vh1 u1, v1;
4 Vh1 lambda;
5 Vh1 p=0;
6
7 fespace Vh2(Th2,P1);
8 Vh2 u2, v2;
Note
It is impossible to define a function just on a part of boundary, so the \(\lambda\) function must be defined on the all domain \(\Omega_1\) such as:
1 Vh1 lambda;
The two Poisson’s problems:
1 problem Pb1 (u1, v1, init=i, solver=Cholesky)
2 = int2d(Th1)(
3 dx(u1)*dx(v1)
4 + dy(u1)*dy(v1)
5 )
6 + int2d(Th1)(
7 - v1
8 )
9 + int1d(Th1, inside)(
10 lambda*v1
11 )
12 + on(outside, u1=0)
13 ;
14
15 problem Pb2 (u2, v2, init=i, solver=Cholesky)
16 = int2d(Th2)(
17 dx(u2)*dx(v2)
18 + dy(u2)*dy(v2)
19 )
20 + int2d(Th2)(
21 - v2
22 )
23 + int1d(Th2, inside)(
24 - lambda*v2
25 )
26 + on(outside, u2=0)
27 ;
And, we define a border matrix, because the \(\lambda\) function is none zero inside the domain \(\Omega_1\):
1 varf b(u2, v2, solver=CG) = int1d(Th1, inside)(u2*v2);
2 matrix B = b(Vh1, Vh1, solver=CG);
The boundary problem function,
1 // Boundary problem function
2 func real[int] BoundaryProblem (real[int] &l){
3 lambda[] = l; //make FE function form l
4 Pb1;
5 Pb2;
6 i++; //no refactorization i != 0
7 v1 = -(u1-u2);
8 lambda[] = B*v1[];
9 return lambda[];
10 }
Note
The difference between the two notations v1
and v1[]
is: v1
is the finite element function and v1[]
is the vector in the canonical basis of the finite element function v1
.
1 // Solve
2 real cpu=clock();
3 LinearCG(BoundaryProblem, p[], eps=1.e-6, nbiter=100);
4 //compute the final solution, because CG works with increment
5 BoundaryProblem(p[]); //solve again to have right u1, u2
6
7 // Display & Plot
8 cout << " -- CPU time schwarz-gc:" << clock()-cpu << endl;
9 plot(u1, u2);