Dual Weighted Reisdual Estimate

In this problem we revisit the Re-entrant Corner Problem but instead of a classical residual estimator we use a dual weighted residual estimator. The aim will be to refine the grid to reduce the error at a given point.

Here we will consider the classic re-entrant corner problem, \begin{align*} -\Delta u &= f, && \text{in } \Omega, \\ u &= g, && \text{on } \partial\Omega, \end{align*} where the domain is given using polar coordinates, \begin{gather*} \Omega = \{ (r,\varphi)\colon r\in(0,1), \varphi\in(0,\Phi) \}~. \end{gather*} For the boundary condition \(g\), we set it to the trace of the function \(u\), given by \begin{gather*} u(r,\varphi) = r^{\frac{\pi}{\Phi}} \sin\big(\frac{\pi}{\Phi} \varphi \big) \end{gather*}

import matplotlib
matplotlib.rc( 'image', cmap='jet' )
import matplotlib.pyplot as pyplot
import numpy
from dune.fem.plotting import plotPointData as plot
import dune.grid as grid
import dune.fem as fem
import dune.common as common
import dune.generator.algorithm as algorithm
from dune.fem.view import adaptiveLeafGridView as adaptiveGridView
from dune.fem.space import lagrange as solutionSpace
from dune.alugrid import aluConformGrid as leafGridView
from ufl import *
from dune.ufl import DirichletBC, expression2GF

# set the angle for the corner (0<angle<=360)
cornerAngle = 320.

# use a second order space
order = 2

We first define the domain and set up the grid and space. We need this twice - once for a computation on a globally refined grid and once for an adaptive one so we put the setup into a function:

We first define the grid for this domain (vertices are the origin and 4 equally spaced points on the unit sphere starting with (1,0) and ending at (cos(cornerAngle), sin(cornerAngle))

Next we define the model together with the exact solution.

def setup():
    vertices = numpy.zeros((8, 2))
    vertices[0] = [0, 0]
    for i in range(0, 7):
        vertices[i+1] = [numpy.cos(cornerAngle/6*numpy.pi/180*i),
    triangles = numpy.array([[2,1,0], [0,3,2], [4,3,0],
                             [0,5,4], [6,5,0], [0,7,6]])
    domain = {"vertices": vertices, "simplices": triangles}
    gridView = adaptiveGridView( leafGridView(domain) )
    space = solutionSpace(gridView, order=order)

    from dune.fem.scheme import galerkin as solutionScheme
    u = TrialFunction(space)
    v = TestFunction(space)
    x = SpatialCoordinate(space.cell())

    # exact solution for this angle
    Phi = cornerAngle / 180 * pi
    phi = atan_2(x[1], x[0]) + conditional(x[1] < 0, 2*pi, 0)
    exact = dot(x, x)**(pi/2/Phi) * sin(pi/Phi * phi)
    a = dot(grad(u), grad(v)) * dx

    # set up the scheme
    laplace = solutionScheme([a==0, DirichletBC(space, exact, 1)], solver="cg",
    uh = space.interpolate(0, name="solution")
    return uh, exact, laplace

uh, exact, laplace = setup()

Now we can setup the functional which will be \(J(v)=v(P)\) where \(P=(0.4,0.4)\) is some point in the computational domain at which we want to minimize the error. To compute the dwr estimator we need the solution to the dual problem with right hand side \(J(\varphi_i)\) for all basis functions \(\varphi_i\). This is not directly expressible in UFL and we therefore implement this using a small C++ function which we then export to Python:

#include <dune/fem/gridpart/common/entitysearch.hh>
#include <dune/fem/function/common/localcontribution.hh>
#include <dune/fem/function/localfunction/const.hh>
#include <dune/fem/common/bindguard.hh>

// Note: the coefficients of the functional will be ADDED to the storage of the functional
template <class Point, class Functional, class Error>
double pointFunctional(const Point &point, Functional &functional, Error &error)
  // first find the entity containing `point`
  typedef typename Functional::DiscreteFunctionSpaceType::GridPartType GridPartType;
  Dune::Fem::EntitySearch<GridPartType> search(functional.space().gridPart());
  const auto &entity = search(point);
  const auto localPoint = entity.geometry().local(point);

  // add the contributions from the basis functions to `functional`
  Dune::Fem::AddLocalContribution< Functional > wLocal( functional );
    auto guard = Dune::Fem::bindGuard( wLocal, entity );
    typename Functional::RangeType one( 1 );
    wLocal.axpy(localPoint, one);

  // in addition we also want to compute the value of `error` at `point`
  auto localError = constLocalFunction(error);
  return localError.evaluate(localPoint)[0];
from dune.fem.scheme import galerkin as solutionScheme
spaceZ = solutionSpace(uh.space.gridView, order=order+1)
u = TrialFunction(spaceZ)
v = TestFunction(spaceZ)
x = SpatialCoordinate(spaceZ)
a = dot(grad(u), grad(v)) * dx
dual = solutionScheme([a==0,DirichletBC(spaceZ,0)], solver="cg")
z = spaceZ.interpolate(0, name="dual")
zh = uh.copy(name="dual_h")
point = common.FieldVector([0.4,0.4])
pointFunctional = z.copy("dual_rhs")
eh = expression2GF( uh.space.gridView, abs(exact-uh), order=order+1 )
computeFunctional = algorithm.load("pointFunctional", "laplace-dwr.hh",
            point, pointFunctional, eh)
from dune.fem.space import finiteVolume as estimatorSpace
from dune.fem.operator import galerkin as estimatorOp

fvspace = estimatorSpace(uh.space.gridView)
estimate = fvspace.interpolate([0], name="estimate")

u = TrialFunction(uh.space.as_ufl())
v = TestFunction(fvspace)
n = FacetNormal(fvspace.cell())
estimator_ufl = abs(div(grad(u)))*abs(z-zh) * v * dx +\
                abs(inner(jump(grad(u)), n('+')))*abs(avg(z-zh)) * avg(v) * dS
estimator = estimatorOp(estimator_ufl)
tolerance = 1e-6

Let us solve over a loop (solve,estimate,mark) and plot the solutions side by side.

h1error = dot(grad(uh - exact), grad(uh - exact))
fig = pyplot.figure(figsize=(10,10))
count = 0
errorVector    = []
estimateVector = []
dofs           = []
while True:
    if count%9 == 8:
        plot(uh, figure=(fig, 131+count//9), colorbar=False)
    error = computeFunctional(point, pointFunctional,eh)
    dual.solve(target=z, rhs=pointFunctional)
    estimator(uh, estimate)
    eta = sum(estimate.dofVector)
    dofs           += [uh.space.size]
    errorVector    += [error]
    estimateVector += [eta]
    if count%3 == 2:
        print(count, ": size=", uh.space.gridView.size(0), "estimate=", eta, "error=", error)
    if eta < tolerance:
    marked = fem.mark(estimate,eta/uh.space.gridView.size(0))
    fem.adapt(uh) # can also be a list or tuple of function to prolong/restrict
    count += 1
plot(uh, figure=(fig, 131+2), colorbar=False)
2 : size= 53 estimate= 0.0018004758731188929 error= 0.0014695307066888708
5 : size= 121 estimate= 0.0004859276474887495 error= 0.0005191292034534833
8 : size= 229 estimate= 0.00015108875865691148 error= 0.00015704482739697934
11 : size= 415 estimate= 5.215284516940018e-05 error= 5.3706338720538316e-05
14 : size= 733 estimate= 1.607527828947397e-05 error= 1.534557471133402e-05
17 : size= 1283 estimate= 5.483626451515096e-06 error= 5.4031600061899e-06
20 : size= 2248 estimate= 1.7508613370287128e-06 error= 1.6480344587899154e-06

Let’s take a close up look of the refined region around the point of interest and the origin:

fig = pyplot.figure(figsize=(30,10))
plot(uh, figure=(fig, 131), xlim=(-0.5, 0.5), ylim=(-0.5, 0.5),
        gridLines="white", colorbar={"shrink": 0.75}, linewidth=2)
plot(uh, figure=(fig, 132), xlim=(-0.1, 0.5), ylim=(-0.1, 0.5),
        gridLines="white", colorbar={"shrink": 0.75}, linewidth=2)
plot(uh, figure=(fig, 133), xlim=(-0.02, 0.5), ylim=(-0.02, 0.5),
        gridLines="white", colorbar={"shrink": 0.75}, linewidth=2)

fig = pyplot.figure(figsize=(30,10))
from dune.fem.function import levelFunction
levels = levelFunction(uh.space.gridView)
plot(levels, figure=(fig, 131), xlim=(-0.5, 0.5), ylim=(-0.5, 0.5),
        gridLines="white", colorbar={"shrink": 0.75})
plot(levels, figure=(fig, 132), xlim=(-0.1, 0.5), ylim=(-0.1, 0.5),
        gridLines="white", colorbar={"shrink": 0.75})
plot(levels, figure=(fig, 133), xlim=(-0.02, 0.5), ylim=(-0.02, 0.5),
        gridLines="white", colorbar={"shrink": 0.75})

This page was generated from the notebook laplace-dwr_nb.ipynb and is part of the tutorial for the dune-fem python bindings DOI