CG2 for poisson equation should be 3rd order L2 convergence but for some reason I am getting near 4th order convergence. If I use DG2 elements, I get the 3rd order convergence as theory suggests. Why is this happening? Below is the snippet of my code mesh = UnitSquareMesh(20,20,quadrilateral=False) Q = FunctionSpace(mesh,CG,2) ... x = SpatialCoordinate(Q.mesh()) u_exact = interpolate(sin(2*x[0]**2*math.pi)*sin(2*x[1]**2*math.pi),Q) f = interpolate( 16*math.pi**2*sin(2*math.pi*x[0]**2)*sin(2*math.pi*x[1]**2 )*(x[0]**2+x[1]**2) - 4*math.pi*(cos(2*math.pi*x[0]**2)*sin(2*math.pi*x[1]**2) + sin(2 *math.pi*x[0]**2)*cos(2*math.pi*x[1]**2)),Q) ... L2_error = errornorm(u_exact,u_h) I never had this "problem" earlier this year when I ran CG2 experiments (though I didn't use the UFL format, but I don't think that's the problem here). Justin
Probably super-convergence due to the regular mesh conforming with symmetries of the exact solution. Either that or Firedrake is magic :) all the best --cjc On 4 October 2017 at 22:07, Justin Chang <jychang48@gmail.com> wrote:
CG2 for poisson equation should be 3rd order L2 convergence but for some reason I am getting near 4th order convergence. If I use DG2 elements, I get the 3rd order convergence as theory suggests.
Why is this happening? Below is the snippet of my code
mesh = UnitSquareMesh(20,20,quadrilateral=False)
Q = FunctionSpace(mesh,CG,2)
...
x = SpatialCoordinate(Q.mesh())
u_exact = interpolate(sin(2*x[0]**2*math.pi)*sin(2*x[1]**2*math.pi),Q)
f = interpolate( 16*math.pi**2*sin(2*math.pi*x[0]**2)*sin(2*math.pi*x[1]** 2)*(x[0]**2+x[1]**2)
- 4*math.pi*(cos(2*math.pi*x[0]**2)*sin(2*math.pi*x[1]**2) + sin(2 *math.pi*x[0]**2)*cos(2*math.pi*x[1]**2)),Q)
...
L2_error = errornorm(u_exact,u_h)
I never had this "problem" earlier this year when I ran CG2 experiments (though I didn't use the UFL format, but I don't think that's the problem here).
Justin
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In your code, you are comparing your solution with the interpolated analytic solution in the same FE space as your approximation. What happens if you interpolate your exact solution into a higher order finite element space, like CG3 or CG4? I suspect you may get the rate you're expecting after that (if not, then some magic is happening...) Best regards, - Thomas ________________________________ From: firedrake-bounces@imperial.ac.uk <firedrake-bounces@imperial.ac.uk> on behalf of Justin Chang <jychang48@gmail.com> Sent: 04 October 2017 22:07:15 To: firedrake Subject: [firedrake] CG2 giving me 4th order L2 convergence CG2 for poisson equation should be 3rd order L2 convergence but for some reason I am getting near 4th order convergence. If I use DG2 elements, I get the 3rd order convergence as theory suggests. Why is this happening? Below is the snippet of my code mesh = UnitSquareMesh(20,20,quadrilateral=False) Q = FunctionSpace(mesh,CG,2) ... x = SpatialCoordinate(Q.mesh()) u_exact = interpolate(sin(2*x[0]**2*math.pi)*sin(2*x[1]**2*math.pi),Q) f = interpolate( 16*math.pi**2*sin(2*math.pi*x[0]**2)*sin(2*math.pi*x[1]**2)*(x[0]**2+x[1]**2) - 4*math.pi*(cos(2*math.pi*x[0]**2)*sin(2*math.pi*x[1]**2) + sin(2*math.pi*x[0]**2)*cos(2*math.pi*x[1]**2)),Q) ... L2_error = errornorm(u_exact,u_h) I never had this "problem" earlier this year when I ran CG2 experiments (though I didn't use the UFL format, but I don't think that's the problem here). Justin
Ah yes, that is precisely the problem. Back then, I think the errornorm() function automatically did some degree_rise=3 or whatever. If I now interpolate the exact solution onto a CG3 or CG4, I do in fact get the convergence rate I would expect. Thanks
participants (3)
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Colin Cotter
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Gibson, Thomas
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Justin Chang