Custom quadrature rule for Quadrature element

1

Hi,
I want to use the custom quadrature rule patch proposed by @bleyerj in https://gitlab.enpc.fr/navier-fenics/fenics-optim/-/blob/bcd1882f34ffbb9e90fb1df50c184d28daab9162/fenics_optim/custom_quadrature_schemes.py with Quadrature element. Nevertheless, the naive approach does not work, see MWE:

import dolfin as df
import matplotlib.pylab as plt
import numpy as np

class ScalarFunction(df.UserExpression):
    def __init__(self, fun_to_eval, **kwargs):
        super().__init__(**kwargs)
        self.fun_to_eval = fun_to_eval

    def eval(self, value, x):
        value[0] = self.fun_to_eval(x)
    
    def value_shape(self):
        return ()

def custom_project(f,V,dx):
    Pf = df.TrialFunction(V)
    v = df.TestFunction(V)
    Pf_h = df.Function(V)
    df.solve(df.inner(Pf,v)*dx==df.inner(f,v)*dx,Pf_h)
    return Pf_h

def fun_to_eval(x):
    
    if x >= 0.3:
        return x**2
    else:
        return x-1.
    
if __name__ == '__main__':
    mesh = df.IntervalMesh(10, 0.0, 1.0)
    x = df.MeshCoordinates(mesh)

    fe_space = 'DG'
    fe_degree = 1

    QUAD_DEG = 2
    dx_deg = df.dx(metadata={"quadrature_degree":QUAD_DEG})
    
    # Function space with DoFs at quadrature points defined as 
    # custom monkey patch by @bleyerj
    QE_quad = df.FiniteElement("Quadrature", mesh.ufl_cell(),
                               degree=QUAD_DEG,
                               quad_scheme="lobatto")
    
    QE = df.FiniteElement(family=fe_space, cell=mesh.ufl_cell(),
                          degree=fe_degree, quad_scheme="lobatto")

    V = df.FunctionSpace(mesh, QE)


    # Interpolate on quadrature element instead of DG:
    ff = ScalarFunction(fun_to_eval, element=QE_quad)

    f = custom_project(ff, V, dx_deg)
    
    xx = np.linspace(0, 1, 50)
    fx = np.zeros(len(xx))
    fx_p = np.zeros(len(xx))
    for i in range(len(xx)):
        fx[i] = f(xx[i])
        fx_p[i] = fun_to_eval(xx[i])
        
    plt.plot(xx, fx_p, label='analytical')
    plt.plot(xx, fx, 'o', label='FE')
    plt.legend()
    plt.show()

Dolfin complains:
Exception: Quadrature element must have specified quadrature scheme (lobatto) equal to the integral (default).

What is the proper way to use a custom quadrature scheme?

BTW: Changing to ‘vertex’ quadrature throws the same Exception.
Dolfin 2019.2, Ubuntu LTS 20.04, PPA

2

Hi,
first I think you have to import the custom_quadrature_schemes module, then you have to specify the “scheme” to the integral measure:

dx_deg = df.dx(metadata={"quadrature_degree": QUAD_DEG}, scheme=scheme)

I hope this helps.

1 Like