Finding error in computing error for stokes equation

[A123-bansal123]

Hello everyone

Below I attached a code of stokes equation with homogeneous boundary condition and exact solution is divergence preserving but It doesn't work properly. The error doesn't decreases as the mesh become finer and finer. Please help me in resolving this error.

from firedrake import *
N = 8
# Define mesh
M = UnitSquareMesh(N, N)
# Define Function Spaces
V = VectorFunctionSpace(M, "Lagrange", 2)
W = FunctionSpace(M, "Lagrange", 1)
R=  FunctionSpace(M, "Real",0)
Z = V * W *R
# Define Test and 
u, p ,c = TrialFunctions(Z)
v, q ,d = TestFunctions(Z)

f = Function(V)
x, y = SpatialCoordinate(M)
source = as_vector((4*pi*pi*sin(2*pi*x),-4*pi*pi*sin(2*pi*y)))
f.interpolate(source)
u_exact = Function(V)
velocity = as_vector((sin(2*pi*y)*(1-cos(2*pi*x)),sin(2*pi*x)*(cos(2*pi*y)-1)))
u_exact.interpolate(velocity)
p_exact = Function(W)
p_exact.interpolate((2*pi*(cos(2*pi*y)-cos(2*pi*x))))
sig_e=grad(u_exact)
a = (inner(grad(u), grad(v)) - inner(p, div(v)) - inner(div(u), q))*dx + p*d*dx + q*c*dx
L = inner(f, v) * dx
 
bc = DirichletBC(Z.sub(0), u_exact , "on_boundary")
z = Function(Z)
solve(a == L, z, bc)
u, p, c = z.subfunctions

error = dot(u-u_exact,u-u_exact)*dx 
E = sqrt(abs(assemble(error)))
print("error =", E)

[ksagiyam]

A quick inspection indicates that the source term should be:

source = as_vector((4 * pi * pi * (-2 * sin(2 * pi * y) * cos(2 * pi * x) - sin(2 * pi * x) + sin(2 * pi * y)),
                    4 * pi * pi * ( 2 * sin(2 * pi * x) * cos(2 * pi * y) - sin(2 * pi * x) + sin(2 * pi * y))))

if u_exact and p_exact are to be kept unchanged.

[A123-bansal123]

Sir, Thank you for your response. your suggestion is correct.
mesh error of u in L2 norm order of convergence
88 0.0031931 -
1616 0.00021056 3.921
3232 1.334e-05 3.98
6464 8.371e-07 3.99

Here, I am getting 4th order of convergence of u in L2 norm but as am approximating my u with polynomial of order 2 so, according to approximation theory, the order of convergence of u in L2 norm should be 3.

why this change occur here ?

[A123-bansal123]

Sir, Thank you for your response. your suggestion is correct. mesh error of u in L2 norm order of convergence ``` mesh error of u in L_2 norm order of convergence 8* *8 0.0031931 -16**16 0.00021056 3.921 32* *32 1.334e-05 3.9864**64 8.371e-07 3.99 ``` Here, I am getting 4th order of convergence of u in L2 norm but as am approximating my u with polynomial of order 2 so, according to approximation theory, the order of convergence of u in L2 norm should be 3. Why does this change occur here ?

…

On Fri, Mar 24, 2023 at 4:59 AM ksagiyam ***@***.***> wrote: A quick inspection indicates that the source term should be: source = as_vector((4 * pi * pi * (-2 * sin(2 * pi * y) * cos(2 * pi * x) - sin(2 * pi * x) + sin(2 * pi * y)), 4 * pi * pi * ( 2 * sin(2 * pi * x) * cos(2 * pi * y) - sin(2 * pi * x) + sin(2 * pi * y)))) if u_exact and p_exact are to be kept unchanged. — Reply to this email directly, view it on GitHub <#2834 (comment)>, or unsubscribe <https://github.com/notifications/unsubscribe-auth/A3SVU4MVYZQRO7NTTFM6JU3W5TMFVANCNFSM6AAAAAAV7TPXMI> . You are receiving this because you authored the thread.Message ID: ***@***.*** com>