 # Periodic boundary about taylor problem

Hi,
I’m trying to solve the taylor vortex problem with periodic boundary in the two-dimensional domain x,yϵ[-1,1]. I use the one order time precision discrete the equation, and Initialize the velocity and pressure with analytic solutions.

The procedure as follows:

mesh Th=square(32,32,[-1+2x,-1+2y]);
fespace Vh(Th,P2,periodic=[[2,y],[4,y],[1,x],[3,x]]);
fespace Ph(Th,P1,periodic=[[2,y],[4,y],[1,x],[3,x]]);
Vh ux,uy,uxold,uyold,vx,vy,dux,duy;
Ph p,pold,q,dp;

real err=10,eps=1e-6,dt=0.02,errt;
uxold=-cos(pix)sin(piy);
uyold=sin(pi
x)cos(piy);
pold=-(cos(2pix)+cos(2piy))/4;
plot([uxold,uyold],wait=1);

solve taylor([ux,uy,p],[vx,vy,q]) =
int2d(Th)(
(uxvx+uyvy)/dt
+(uxolddx(ux)+uyolddy(ux))vx
+(uxold
dx(uy)+uyolddy(uy))vy
-p
(dx(vx)+dy(vy))
-q
(dx(ux)+dy(uy))
)
-int2d(Th)((uxoldvx+uyoldvy)/dt)
;

When running the case, FreeFem++ show that :

Problem build of FEspace (2d) (may be: due to periodic boundary condition missing) FH
The number of DF must be 9216 and it is 9539

I don’t know why, I hope somebody could have a look and give me some advices I am having exactly the same problem (same error statement) for the NSE in 3D. It definitely is the issue with periodic spaces; if you use fespace Vh(Th,P2) and fespace Vh(Th,P1), the error message will disappear. Hopefully someone will answer you and it will help me too:)