Thank you to the developers of the ngsolve program for providing such a valuable learning platform. Due to the presence of time derivatives in the boundary terms of the equation, it is necessary to implement dtref at the boundaries. However, an error was encountered in doing so：Operator “dt” does not exist for CompoundFESpace!，Despite various attempts, the endeavor still ended in failure.I am very grateful for the guidance provided by the ngsolve team members.
main equation:
main code：
from netgen.geom2d import SplineGeometry
from ngsolve import *
from xfem import *
import numpy as np
Space finite element order
order = 1
Time finite element order
k_t = 2
Final simulation time
tend = 1.0
Time step
sizeh=4
PDE parameter
Diff_CF=0.02
Delta=0.02
Sigma=1
Diff_Gamma_CF=0.02
Delta_Gamma=0.02
k=1
delta_t = 1 / 2**(sizeh)
T_time=np.arange(0,1+delta_t ,delta_t )
geo = SplineGeometry()
geo.AddRectangle( (0, 0), (1, 1), bc = “outer”)
mesh = Mesh( geo.GenerateMesh(maxh=1/2**sizeh))
Normal=specialcf.normal(mesh.dim)
TLen=len(T_time)
def dt(u):
return 1.0 / delta_t * dtref(u)
told = Parameter(0)
t = told + delta_t * tref
dxt = delta_t * dxtref(mesh, time_order=8)
dst=delta_t*dxtref(mesh,time_order=8,definedon=‘mesh.Bounderies(outer)’)
Fan_Space=H1(mesh,order=1)
Mu_Space=H1(mesh,order=1)
Mu_Gamma_Space=Compress(H1(mesh,order=1,definedon=mesh.Boundaries(‘outer’)))
tfe = ScalarTimeFE(k_t) # type: ignore
ST_Fan=tfeFan_Space
ST_Mu=tfeMu_Space
ST_Mu_Gamma=tfe*Mu_Gamma_Space
fes=ST_FanST_MuST_Mu_Gamma
(Fan,Mu,Mu_Gamma),(theta1,theta2,theta3)=fes.TnT()
mass_Fan=np.zeros(TLen-1)
mass_Fan_Gamma=np.zeros(TLen-1)
H1_Err_Phi=np.zeros(TLen-1)
Err_Phi=np.zeros(TLen-1)
gfu_iter=GridFunction(fes)
Init_Fan=GridFunction(Fan_Space)
potential function second derive
def Doub_second(u):
return 3*u**2-1
def Doub_First(u):
return u**3-u
a_Lin=BilinearForm(fes)
the fan not the origin fan fan1=fan^{n-1}+(t-t_{n-1})fan
a_Lin+=Fantheta1dxt+delta_ttrefdt(Fan)theta1dxt+Diff_CFgrad(Mu)grad(theta1)dxt
+Fan.Trace()theta3dst+delta_ttrefdt(Fan.Trace())theta3dst#+Diff_Gamma_CFgrad(Mu_Gamma).Trace()grad(theta3).Trace()dst
#-Mutheta2dxt-Mu_Gammatheta2.Trace()dst
#+delta_ttrefDeltaSigmagrad(Fan)grad(theta2)dxt+delta_ttrefDelta_Gammakgrad(Fan).Trace()grad(theta2).Trace()dst
#+1/DeltaSigmadelta_ttrefDoub_second(Init_Fan+delta_ttrefgfu_iter.components[0])Fantheta2dxt
#+1/Delta_Gamma**delta_ttrefDoub_second(Init_Fan+delta_ttrefgfu_iter.components[0])Fantheta2dst
Best regards
ChLun Ye