### Question with enriching basis function near boundary

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10 weeks ago by
Dear friends,

I am doing some simulation with fluid. The resolution of boundary layer is important for solution. In order to improve the accuracy, I want to enrich the basis function near boundary with some special exponential function. I wonder whether this is possible to implement with FEniCS. Hope you reply soon. Thank you for your help all the time.

Yours sincerely,
Qiming Zhu
Community: FEniCS Project

### 1 Answer

1
10 weeks ago by
There was some work a while back on partition of unity methods for adding enrichment functions into FEniCS,

https://repository.tudelft.nl/islandora/object/uuid:49e4dcc2-d22c-429e-a70d-73c30d72ba4a

but it looks pretty involved, with modifications of DOLFIN and FFC that I don't think were ever merged back into the main project.  (Maybe someone with more knowledge can correct me here.)

On the other hand, if there are only a few "special" functions for the whole problem, you could try to jury-rig something with "Real" type elements, e.g.,

from dolfin import *

# You need really good quadrature to fully resolve the special exponential
# function.
dx = dx(metadata={"quadrature_degree":20})

mesh = UnitIntervalMesh(5)
VE = FiniteElement("Lagrange",mesh.ufl_cell(),1)
RE = FiniteElement("Real",mesh.ufl_cell(),0)
V = FunctionSpace(mesh,MixedElement([VE,RE]))
V_unenriched = FunctionSpace(mesh,VE)

u_mixed = TrialFunction(V)
v_mixed = TestFunction(V)

# advection--diffusion parameters
kappa = Constant(0.01)
u_adv = Constant((1.0,))

# Peclet number
Pe = sqrt(inner(u_adv,u_adv))/kappa

# Define a special function to get an accurate boundary layer
x = SpatialCoordinate(mesh)[0]
specialFunction = exp(Pe*x) - (1.0-x) - x*exp(Pe)

# Helper functions to extract parts from elements of the mixed function space
def linearPart(u_mixed):
u_linear,_ = split(u_mixed)
return u_linear
def specialPart(u_mixed):
_,u_specialCoeff = split(u_mixed)
return u_specialCoeff*specialFunction
def mixedToFunc(u_mixed):
return linearPart(u_mixed) + specialPart(u_mixed)

# What to plug into the variational problem
u = mixedToFunc(u_mixed)
v = mixedToFunc(v_mixed)

# Define variational forms for advection--diffusion (with no stabilization)
def a(u,v):
return kappa*inner(grad(u),grad(v))*dx + inner(grad(u),u_adv)*v*dx
def L(v):
return Constant(0.0)*v*dx

lhs = a(u,v)
rhs = L(v)

def getBCs(V):
bc_left = DirichletBC(V, 0.0,"on_boundary && near(x[0],0.0)")
bc_right = DirichletBC(V, 1.0,"on_boundary && near(x[0],1.0)")
return [bc_left, bc_right]

# Solve for enriched approximate solution
u_mixed = Function(V)
solve(lhs==rhs,u_mixed,getBCs(V.sub(0)))

# Solve using the linear space only, for comparison:
u_unenriched = TrialFunction(V_unenriched)
v_unenriched = TestFunction(V_unenriched)
lhs = a(u_unenriched,v_unenriched)
rhs = L(v_unenriched)
u_unenriched = Function(V_unenriched)
solve(lhs==rhs,u_unenriched,getBCs(V_unenriched))

# Refined mesh for plotting
refinedMesh = UnitIntervalMesh(1000)
V_refined = FunctionSpace(refinedMesh,"Lagrange",1)

# Plot results, projected to the refined mesh:
import matplotlib.pyplot as plt
plot(project(mixedToFunc(u_mixed),V_refined))
plot(project(u_unenriched,V_refined))
#plot(project(specialPart(u_mixed),V_refined))
#plot(project(linearPart(u_mixed),V_refined))
plt.autoscale()
plt.show()
​
Thank you for your detailed reply very much. I appreciate your help all the time. I would try to implement my problem with your brilliant idea and "Real" type element. Thank you for your help.
written 10 weeks ago by zqm1992
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