references

Author: pbrubeck

demos/adaptive_multigrid/adaptive_convergence.png

@@ -51,9 +51,9 @@ Our approach strongly follows the similar problem in this `lecture course <https
 
    def solve_poisson(mesh, params):
       V = FunctionSpace(mesh, "CG", 1)
-      uh = Function(V, name="Solution")
+      uh = Function(V, name="solution")
       v = TestFunction(V)
-      bc = DirichletBC(V, Constant(0), "on_boundary")
+      bc = DirichletBC(V, 0, "on_boundary")
       f = Constant(1)
       F = inner(grad(uh), grad(v))*dx - inner(f, v)*dx
 
@@ -146,16 +146,16 @@ Our approach will be to compute the estimator over all elements and selectively
            error_est = eta_.norm()
        return eta, error_est
 
-The next step is to choose which elements to refine. For this we use Dörfler marking [Dörfler1996]:  
+The next step is to choose which elements to refine. For this we use Dörfler marking :cite:`Dorfler1996`:
 
 .. math::
    \eta_K \geq \theta \text{max}_L \eta_L
 
-The logic is to select an element :math:`K` to refine if the estimator is greater than some factor :math:`\theta` of the maximum error estimate of the mesh, where :math:`\theta` ranges from 0 to 1. In our code we choose :math:`theta=0.5`.
+The logic is to select an element :math:`K` to refine if the estimator is greater than some factor :math:`\theta` of the maximum error estimate of the mesh, where :math:`\theta` ranges from 0 to 1. In our code we choose :math:`\theta=0.5`.
 With these helper functions complete, we can solve the system iteratively. In the max_iterations is the number of total levels we want to perform multigrid on. We will solve for 15 levels. At every level :math:`l`, we first compute the solution using multigrid with patch relaxation up till level :math:`l`. We then use the current approximation of the solution to estimate the error across the mesh. Finally, we refine the mesh and repeat. ::
 
    theta = 0.5
-   max_iterations = 10
+   max_iterations = 15
    est_errors = []
    dofs = []
    for i in range(max_iterations):
@@ -187,10 +187,10 @@ With these helper functions complete, we can solve the system iteratively. In th
    plt.ylabel("Error estimate of the energy norm")
    plt.xlabel("Number of degrees of freedom")
    plt.legend()
-   plt.savefig(f"output/adaptive_convergence_{max_iterations-1}.png")
+   plt.savefig("output/adaptive_convergence.png")
 
 
-To perform Dörfler marking, refine the current mesh, and add the mesh to the :class:`.AdaptiveMeshHierarchy`, we use the ``amh.adapt(eta, theta)`` method. In this method the input is the recently computed error estimator :math:`eta` and the Dörfler marking parameter :math:`theta`. The method always performs this on the current fine mesh in the hierarchy. There is another method for adding a mesh to the hierarchy: ``amh.add_mesh(mesh)``. In this method, refinement on the mesh is performed externally by some custom procedure and the resulting mesh directly gets added to the hierarchy.
+To perform Dörfler marking, refine the current mesh, and add the mesh to the :class:`.AdaptiveMeshHierarchy`, we use the ``amh.adapt(eta, theta)`` method. In this method the input is the recently computed error estimator ``eta`` and the Dörfler marking parameter ``theta``. The method always performs this on the current fine mesh in the hierarchy. There is another method for adding a mesh to the hierarchy: ``amh.add_mesh(mesh)``. In this method, refinement on the mesh is performed externally by some custom procedure and the resulting mesh directly gets added to the hierarchy.
 The meshes now refine according to the error estimator. The error estimators at levels 3,5, and 15 are shown below. Zooming into the vertex of the L-shape at level 15 shows the error indicator remains strongest there. Further refinements will focus on that area.
 
 +-------------------------------+-------------------------------+-------------------------------+
@@ -216,11 +216,14 @@ The solutions at level 4 and 15 are shown below.
 
 The convergence follows the expected optimal behavior:
 
-.. figure:: adaptive_convergence_9.png
+.. figure:: adaptive_convergence.png
    :align: center
    :alt: Convergence of the error estimator.
 
-References
-----------
-[Dörfler1996] W. Dörfler. A convergent adaptive algorithm for poisson’s equation. SIAM Journal on Numerical Analysis, 33(3):1106–1124, 1996.
 
+A runnable python version of this demo can be found :demo:`here<adaptive_multigrid.py>`.
+
+.. rubric:: References
+
+.. bibliography:: demo_references.bib
+   :filter: docname in docnames

demos/adaptive_multigrid/adaptive_convergence_9.png

@@ -380,3 +380,14 @@ @Article{Farrell2015
   pages =        {A2026--A2045},
   doi =          {10.1137/140984798},
 }
+
+@article{Dorfler1996,
+  title={A convergent adaptive algorithm for Poisson’s equation},
+  author={D{\"o}rfler, Willy},
+  journal={SIAM Journal on Numerical Analysis},
+  volume={33},
+  number={3},
+  pages={1106--1124},
+  year={1996},
+  publisher={SIAM}
+}