The saddle point system of the mixed finite element

General Discussion
1

hello there,

I’m trying to get the underlying linear system of the mixed finite formulation of the poisson problem. The one of the form
Screenshot from 2024-07-16 14-31-21
is there a way to do so in freefem.

2

Yes, there is a way to do so in FreeFEM.

1 Like
3

Thank you for your reply,

Can you please provide a refernce or an example?

4

Here is one FreeFem-sources/examples/tutorial/LaplaceRT.edp at master · FreeFem/FreeFem-sources · GitHub.

5

I have seen this example in the documentation, but I’d like to get access to the submatrices M and B I have tried the following

// Mesh
mesh Th = square(10, 10);
func PkV = RT0;
func PkP = P0;
func Pk = [PkV, PkP];               

fespace Wh(Th, Pk);           // local finite element space


varf vMixedLaplace([u1, u2, p], [v1, v2, q]) = int2d(Th)(u1 * v1 + u2 * v2 + p*(dx(v1) + dy(v2)) + (dx(u1) + dy(u2))*q)
                                             + on(4, u1 = 1.0, u2 = 1.0);
matrix A;
A = vMixedLaplace(Wh, Wh, tgv = -1);

but the resulting matrix A does not have the form of the saddle point problem.

6

also, I tried to get the matrices M and B and solve the systen using the Schur complement, but I get wrong answer

// Fespace
fespace Vh(Th, RT0);
Vh [u1, u2];
Vh [v1, v2];

fespace Ph(Th, P0);
Ph p, q;

varf m([u1, u2], [v1, v2]) = int2d(Th)(u1*v1+u2*v2)+ on(4, u1=1, u2=1);
matrix M=m(Vh,Vh,tgv = -1);

varf b1([p],[v1,v2]) = int2d(Th)(p*dx(v1)+p*dy(v2));
matrix B=b1(Ph,Vh);

varf l1([unused],[v1,v2]) = int1d(Th, 1, 2, 3)(gd*v1*N.x+gd*v2*N.y,tgv = -1);
real[int] r1 = l1(0,Vh);

varf l2(unused, q) = int2d(Th)(-q);
real[int] r2 = l2(0,Ph);
7

Here is one example solving the system using the Schur complement: FreeFem-sources/examples/hpddm/laplace-RT-2d-PETSc.edp at master · FreeFem/FreeFem-sources · GitHub.