THE exact solution between its interpolate error is lager

mathematics
1 
from dolfin import *
import numpy as np
from math import *
import os, sys
import sympy as sp
from sympy import exp, sin, pi, cos
T = 1          # final time

num_steps = 100    # number of time steps
dt = T / num_steps # time step size

mu=0.1

# define symbol expressions of u,p,f using sympyw23 
x,y,t = sp.symbols("x[0],x[1],t")
u1=pi*sin(t)*sin(2*pi*y)*sin(pi*x)*sin(pi*x)

u2=-pi*sin(t)*sin(2*pi*x)*sin(pi*y)*sin(pi*y)
ps=sin(t)*sin(pi*x)*sin(pi*y)
f1= -mu*sp.diff(u1,x,2) - mu*sp.diff(u1,y,2) + sp.diff(ps,x,1)+\
    sp.diff(u1,t,1) + u1*sp.diff(u1,x,1)+u2*sp.diff(u1,y,1)
f2= -mu*sp.diff(u2,x,2) - mu*sp.diff(u2,y,2) + sp.diff(ps,y,1)+\
    sp.diff(u2,t,1) + u1*sp.diff(u2,x,1)+u2*sp.diff(u2,y,1)
# transform the expressions to ccode style
u1 = sp.printing.ccode(u1)
u2 = sp.printing.ccode(u2)
ps = sp.printing.ccode(ps)
f1 = sp.printing.ccode(f1)
f2 = sp.printing.ccode(f2)
# Create mesh and define function spaces
mesh = RectangleMesh(Point(-1, -1), Point(1, 1), 16, 16)
P1 = FiniteElement("Lagrange", mesh.ufl_cell(), 2)
Q = FunctionSpace(mesh, P1) 
#V= FunctionSpace(mesh, "RT", 1) 
V = VectorFunctionSpace(mesh, 'P', 2)
u_e = Expression((u1,u2), degree=5,t=0)
# Define boundaries
def Dirichlt_boundary(x,on_boundary):
    return x[0] < -1+DOLFIN_EPS or x[0] > 1-DOLFIN_EPS or x[1] > 1.0 - DOLFIN_EPS or x[1] < -1+DOLFIN_EPS 
bcu_noslip  = DirichletBC(V, u_e,  Dirichlt_boundary)
p_e=Expression(ps, degree=5,t=0)
t = 0 
for n in range(num_steps): 
    # Update current time 
    t += dt 
    # Compute error 
    u_e.t = t 
    p_e.t = t
    u_d = interpolate(u_e, V) 
    p_d = interpolate(p_e, Q) 
    errorU=errornorm(u_e, u_d, "L2") 
    errorP=errornorm(p_e, p_d, "L2") 
    print('t = %.2f: errorU = %.3g' % (t, errorU))
    print('t = %.2f: errorP = %.3g' % (t, errorP))

I creat an exact solution, and every time I interpolate the sol to the finite element space and get the error between them, it is so bad the error perform.
微信图片_20201214211040
so can anyone explain this thing?

2

I am not sure why you mean that this is a large error? You are using second order polynomials to approximate a series of sin functions.
The error is time dependent as the amplitude of your problem is time dependent, thus the error changes with the amplitude (change your end time to for instance half a period (T=np.pi), to see that the error decreases at the end of the cycle, when the function has a smaller amplitude.

Please, for next time, remove statements that are not necessary to reproduce your code, such as f1 and f2, as well as the bc_noslip and corresponding functions.

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