Set max doc length to 75 for python/dolfinx/ and python/demo/

python/demo/demo_axis.py

@@ -154,8 +154,8 @@ def generate_mesh_sphere_axis(
 # \cdot (\nabla \times \bar{\mathbf{v}})+\varepsilon_{r} k_{0}^{2}
 # \mathbf{E}_s \cdot \bar{\mathbf{v}}+k_{0}^{2}\left(\varepsilon_{r}
 # -\varepsilon_b\right)\mathbf{E}_b \cdot \bar{\mathbf{v}}~\mathrm{d} x\\
-# +\int_{\Omega_{pml}}\left[\boldsymbol{\mu}^{-1}_{pml} \nabla \times \mathbf{E}_s
-# \right]\cdot \nabla \times \bar{\mathbf{v}}-k_{0}^{2}
+# +\int_{\Omega_{pml}}\left[\boldsymbol{\mu}^{-1}_{pml} \nabla \times
+# \mathbf{E}_s \right]\cdot \nabla \times \bar{\mathbf{v}}-k_{0}^{2}
 # \left[\boldsymbol{\varepsilon}_{pml} \mathbf{E}_s \right]\cdot
 # \bar{\mathbf{v}}~ d x=0
 # \end{split}
@@ -172,8 +172,8 @@ def generate_mesh_sphere_axis(
 # -\varepsilon_b\right)\mathbf{E}_b \cdot
 # \bar{\mathbf{v}}~ \rho d\rho dz d \phi\\
 # +\int_{\Omega_{cs}}
-# \int_{0}^{2\pi}\left[\boldsymbol{\mu}^{-1}_{pml} \nabla \times \mathbf{E}_s
-# \right]\cdot \nabla \times \bar{\mathbf{v}}-k_{0}^{2}
+# \int_{0}^{2\pi}\left[\boldsymbol{\mu}^{-1}_{pml} \nabla \times
+# \mathbf{E}_s \right]\cdot \nabla \times \bar{\mathbf{v}}-k_{0}^{2}
 # \left[\boldsymbol{\varepsilon}_{pml} \mathbf{E}_s \right]\cdot
 # \bar{\mathbf{v}}~ \rho d\rho dz d \phi=0
 # \end{split}
@@ -184,8 +184,10 @@ def generate_mesh_sphere_axis(
 #
 # $$
 # \begin{align}
-# \mathbf{E}_s(\rho, z, \phi) &= \sum_m\mathbf{E}^{(m)}_s(\rho, z)e^{-jm\phi} \\
-# \mathbf{E}_b(\rho, z, \phi) &= \sum_m\mathbf{E}^{(m)}_b(\rho, z)e^{-jm\phi} \\
+# \mathbf{E}_s(\rho, z, \phi) &= \sum_m\mathbf{E}^{(m)}_s(\rho, z)
+#   e^{-jm\phi} \\
+# \mathbf{E}_b(\rho, z, \phi) &= \sum_m\mathbf{E}^{(m)}_b(\rho, z)
+#   e^{-jm\phi} \\
 # \bar{\mathbf{v}}(\rho, z, \phi) &=
 # \sum_m\bar{\mathbf{v}}^{(m)}(\rho, z)e^{+jm\phi}
 # \end{align}
@@ -748,8 +750,10 @@ def create_eps_mu(pml, rho, eps_bkg, mu_bkg):
 # m = np.sqrt(eps_au)/n_bkg
 # x = 2*np.pi*radius_sph/wl0*n_bkg
 #
-# q_sca_analyt, q_abs_analyt = scattnlay(np.array([x], dtype=np.complex128),
-#                                        np.array([m], dtype=np.complex128))[2:4]
+# q_sca_analyt, q_abs_analyt = scattnlay(
+#   np.array([x], dtype=np.complex128),
+#   np.array([m], dtype=np.complex128)
+# )[2:4]
 # ```
 #
 # The numerical values are reported here below:

python/demo/demo_biharmonic.py

@@ -33,23 +33,23 @@
 # \nabla^{4} u = f \quad {\rm in} \ \Omega,
 # $$
 #
-# where $\nabla^{4} \equiv \nabla^{2} \nabla^{2}$ is the biharmonic operator
-# and $f$ is a prescribed source term.
+# where $\nabla^{4} \equiv \nabla^{2} \nabla^{2}$ is the biharmonic
+# operator and $f$ is a prescribed source term.
 # To formulate a complete boundary value problem, the biharmonic equation
 # must be complemented by suitable boundary conditions.
 #
 # ### Weak formulation
 #
 # Multiplying the biharmonic equation by a test function and integrating
-# by parts twice leads to a problem of second-order derivatives, which would
-# require $H^{2}$ conforming (roughly $C^{1}$ continuous) basis functions.
-# To solve the biharmonic equation using Lagrange finite element basis
-# functions, the biharmonic equation can be split into two second-order
-# equations (see the Mixed Poisson demo for a mixed method for the Poisson
-# equation), or a variational formulation can be constructed that imposes
-# weak continuity of normal derivatives between finite element cells.
-# This demo uses a discontinuous Galerkin approach to impose continuity
-# of the normal derivative weakly.
+# by parts twice leads to a problem of second-order derivatives, which
+# would require $H^{2}$ conforming (roughly $C^{1}$ continuous) basis
+# functions. To solve the biharmonic equation using Lagrange finite element
+# basis functions, the biharmonic equation can be split into two second-
+# order equations (see the Mixed Poisson demo for a mixed method for the
+# Poisson equation), or a variational formulation can be constructed that
+# imposes weak continuity of normal derivatives between finite element
+# cells. This demo uses a discontinuous Galerkin approach to impose
+# continuity of the normal derivative weakly.
 #
 # Consider a triangulation $\mathcal{T}$ of the domain $\Omega$, where
 # the set of interior facets is denoted by $\mathcal{E}_h^{\rm int}$.
@@ -71,7 +71,8 @@
 # \end{align}
 # $$
 #
-# a weak formulation of the biharmonic problem reads: find $u \in V$ such that
+# a weak formulation of the biharmonic problem reads: find $u \in V$ such
+# that
 #
 # $$
 # a(u,v)=L(v) \quad \forall \ v \in V,
@@ -158,8 +159,9 @@
 # Next, we locate the mesh facets that lie on the boundary
 # $\Gamma_D = \partial\Omega$.
 # We do this using using {py:func}`exterior_facet_indices
-# <dolfinx.mesh.exterior_facet_indices>` which returns all mesh boundary facets
-# (Note: if we are only interested in a subset of those, consider {py:func}`locate_entities_boundary
+# <dolfinx.mesh.exterior_facet_indices>` which returns all mesh boundary
+# facets (Note: if we are only interested in a subset of those, consider
+# {py:func}`locate_entities_boundary
 # <dolfinx.mesh.locate_entities_boundary>`).
 
 tdim = msh.topology.dim
@@ -182,8 +184,8 @@
 
 # Next, we express the variational problem using UFL.
 #
-# First, the penalty parameter $\alpha$ is defined. In addition, we define a
-# variable `h` for the cell diameter $h_E$, a variable `n`for the
+# First, the penalty parameter $\alpha$ is defined. In addition, we define
+# a variable `h` for the cell diameter $h_E$, a variable `n`for the
 # outward-facing normal vector $n$ and a variable `h_avg` for the
 # average size of cells sharing a facet
 # $\left< h \right> = \frac{1}{2} (h_{+} + h_{-})$. Here, the UFL syntax
@@ -195,12 +197,12 @@
 n = ufl.FacetNormal(msh)
 h_avg = (h("+") + h("-")) / 2.0
 
-# After that, we can define the variational problem consisting of the bilinear
-# form $a$ and the linear form $L$. The source term is prescribed as
-# $f = 4.0 \pi^4\sin(\pi x)\sin(\pi y)$. Note that with `dS`, integration is
-# carried out over all the interior facets $\mathcal{E}_h^{\rm int}$, whereas
-# with `ds` it would be only the facets on the boundary of the domain, i.e.
-# $\partial\Omega$. The jump operator
+# After that, we can define the variational problem consisting of the
+# bilinear form $a$ and the linear form $L$. The source term is prescribed
+# as $f = 4.0 \pi^4\sin(\pi x)\sin(\pi y)$. Note that with `dS`,
+# integration is carried out over all the interior facets
+# $\mathcal{E}_h^{\rm int}$, whereas with `ds` it would be only the facets
+# on the boundary of the domain, i.e. $\partial\Omega$. The jump operator
 # $[\!\![ w ]\!\!] = w_{+} \cdot n_{+} + w_{-} \cdot n_{-}$ w.r.t. the
 # outward-facing normal vector $n$ is in UFL available as `jump(w, n)`.
 

python/demo/demo_cahn-hilliard.py

@@ -62,7 +62,8 @@
 # \begin{align}
 # \frac{\partial c}{\partial t} - \nabla \cdot M \nabla\mu
 #     &= 0 \quad {\rm in} \ \Omega, \\
-# \mu -  \frac{d f}{d c} + \lambda \nabla^{2}c &= 0 \quad {\rm in} \ \Omega.
+# \mu -  \frac{d f}{d c} + \lambda \nabla^{2}c &= 0 \quad {\rm in}
+#   \ \Omega.
 # \end{align}
 # $$
 #
@@ -74,8 +75,8 @@
 # \int_{\Omega} \frac{\partial c}{\partial t} q \, {\rm d} x +
 #     \int_{\Omega} M \nabla\mu \cdot \nabla q \, {\rm d} x
 #     &= 0 \quad \forall \ q \in V,  \\
-# \int_{\Omega} \mu v \, {\rm d} x - \int_{\Omega} \frac{d f}{d c} v \, {\rm d} x
-#   - \int_{\Omega} \lambda \nabla c \cdot \nabla v \, {\rm d} x
+# \int_{\Omega} \mu v \, {\rm d} x - \int_{\Omega} \frac{d f}{d c} v \,
+#   {\rm d} x - \int_{\Omega} \lambda \nabla c \cdot \nabla v \, {\rm d} x
 #    &= 0 \quad \forall \ v \in V.
 # \end{align}
 # $$
@@ -89,15 +90,16 @@
 # $$
 # \begin{align}
 # \int_{\Omega} \frac{c_{n+1} - c_{n}}{dt} q \, {\rm d} x
-# + \int_{\Omega} M \nabla \mu_{n+\theta} \cdot \nabla q \, {\rm d} x
-#        &= 0 \quad \forall \ q \in V  \\
-# \int_{\Omega} \mu_{n+1} v  \, {\rm d} x - \int_{\Omega} \frac{d f_{n+1}}{d c} v  \, {\rm d} x
-# - \int_{\Omega} \lambda \nabla c_{n+1} \cdot \nabla v \, {\rm d} x
-#        &= 0 \quad \forall \ v \in V
+#   + \int_{\Omega} M \nabla \mu_{n+\theta} \cdot \nabla q \, {\rm d} x
+#   &= 0 \quad \forall \ q \in V  \\
+# \int_{\Omega} \mu_{n+1} v  \, {\rm d} x - \int_{\Omega}
+#   \frac{d f_{n+1}}{d c} v  \, {\rm d} x - \int_{\Omega} \lambda \nabla
+#   c_{n+1} \cdot \nabla v \, {\rm d} x &= 0 \quad \forall \ v \in V
 # \end{align}
 # $$
 #
-# where $dt = t_{n+1} - t_{n}$ and $\mu_{n+\theta} = (1-\theta) \mu_{n} + \theta \mu_{n+1}$.
+# where $dt = t_{n+1} - t_{n}$ and $\mu_{n+\theta} = (1-\theta) \mu_{n} +
+# \theta \mu_{n+1}$.
 # The task is: given $c_{n}$ and $\mu_{n}$, solve the above equation to
 # find $c_{n+1}$ and $\mu_{n+1}$.
 #

python/demo/demo_hdg.py

@@ -103,8 +103,8 @@ def u_e(x):
 facets = np.arange(num_facets, dtype=np.int32)
 
 # Create the sub-mesh
-# NOTE Despite all facets being present in the submesh, the entity map isn't
-# necessarily the identity in parallel
+# NOTE Despite all facets being present in the submesh, the entity map
+# isn't necessarily the identity in parallel
 facet_mesh, facet_mesh_to_mesh = mesh.create_submesh(msh, fdim, facets)[:2]
 
 # Define function spaces

python/demo/demo_lagrange_variants.py

@@ -37,8 +37,9 @@
 # influence of interpolation point position, we create a degree 10
 # element on an interval using equally spaced points, and plot the basis
 # functions. We create this element using `basix.ufl`'s
-# `element` function. The function `element.tabulate` returns a 3-dimensional
-# array with shape (derivatives, points, (value size) * (basis functions)).
+# `element` function. The function `element.tabulate` returns a 3-
+# dimensional array with shape (derivatives, points, (value size) *
+# (basis functions)).
 # In this example, we only tabulate the 0th derivative and the value
 # size is 1, so we take the slice `[0, :, :]` to get a 2-dimensional
 # array.
@@ -158,7 +159,8 @@ def saw_tooth(x):
 # Lagrange compared to the GLL variant. To quantify the error, we
 # compute the interpolation error in the $L_2$ norm,
 #
-# $$\left\|u - u_h\right\|_2 = \left(\int_0^1 (u - u_h)^2\right)^{\frac{1}{2}},$$
+# $$\left\|u - u_h\right\|_2 =
+#   \left(\int_0^1 (u - u_h)^2\right)^{\frac{1}{2}},$$
 #
 # where $u$ is the function and $u_h$ is its interpolation in the finite
 # element space. The following code uses UFL to compute the $L_2$ error

python/demo/demo_mixed-poisson.py

@@ -8,7 +8,7 @@
 #       jupytext_version: 1.14.1
 # ---
 
-# # Mixed formulation of the Poisson equation with a block-preconditioner/solver
+# # Mixed formulation of the Poisson equation with a block-preconditioner/solver # noqa
 #
 # This demo illustrates how to solve the Poisson equation using a mixed
 # (two-field) formulation and a block-preconditioned iterative solver.

python/demo/demo_mixed-topology.py

@@ -110,8 +110,8 @@
 V_cpp = _cpp.fem.FunctionSpace_float64(mesh, elements_cpp, dofmaps)
 
 # Create forms for each cell type.
-# FIXME This hack is required at the moment because UFL does not yet know about
-# mixed topology meshes.
+# FIXME This hack is required at the moment because UFL does not yet know
+# about mixed topology meshes.
 a = []
 L = []
 for i, cell_name in enumerate(["hexahedron", "prism"]):

python/demo/demo_navier-stokes.py

@@ -8,7 +8,7 @@
 #       jupytext_version: 1.13.6
 # ---
 
-# # Divergence conforming discontinuous Galerkin method for the Navier--Stokes equations
+# # Divergence conforming discontinuous Galerkin method for the Navier--Stokes equations # noqa
 #
 # This demo ({download}`demo_navier-stokes.py`) illustrates how to
 # implement a divergence conforming discontinuous Galerkin method for
@@ -29,7 +29,8 @@
 #
 # $$
 # \begin{align}
-# \partial_t u - \nu \Delta u + (u \cdot \nabla)u + \nabla p &= f \text{ in } \Omega_t, \\
+# \partial_t u - \nu \Delta u + (u \cdot \nabla)u + \nabla p &= f
+# \text{ in } \Omega_t, \\
 # \nabla \cdot u &= 0 \text{ in } \Omega_t,
 # \end{align}
 # $$
@@ -124,16 +125,17 @@
 #     \end{cases}
 # $$
 #
-# where $\Gamma^\psi = \left\{x \in \Gamma; \; \psi(x) \cdot n(x) < 0\right\}$.
+# where $\Gamma^\psi = \left\{x \in \Gamma; \; \psi(x) \cdot n(x) < 0
+# \right\}$.
 #
 # The semi-discrete version problem (in dimensionless form) is: find
 # $(u_h, p_h) \in V_h^{u_D} \times Q_h$ such that
 #
 # $$
 # \begin{align}
 #   \int_\Omega \partial_t u_h \cdot v + a_h(u_h, v_h) + c_h(u_h; u_h, v_h)
-#   + b_h(v_h, p_h) &= \int_\Omega f \cdot v_h + L_{a_h}(v_h) + L_{c_h}(v_h)
-#   \quad \forall v_h \in V_h^0, \\
+#   + b_h(v_h, p_h) &= \int_\Omega f \cdot v_h + L_{a_h}(v_h) +
+#   L_{c_h}(v_h) \quad \forall v_h \in V_h^0, \\
 #   b_h(u_h, q_h) &= 0 \quad \forall q_h \in Q_h,
 # \end{align}
 # $$
@@ -151,8 +153,10 @@
 #   c_h(w; u, v) &= - \sumK \int_K u \cdot \nabla \cdot (v \otimes w)
 #   + \sumK \int_{\partial_K} w \cdot n \hat{u}^{w} \cdot v, \\
 #   L_{a_h}(v_h) &= Re^{-1} \left(- \int_{\partial \Omega} u_D \otimes n :
-#   \nabla_h v_h + \frac{\alpha}{h} u_D \otimes n : v_h \otimes n \right), \\
-#   L_{c_h}(v_h) &= - \int_{\partial \Omega} u_D \cdot n \hat{u}_D \cdot v_h, \\
+#   \nabla_h v_h + \frac{\alpha}{h} u_D \otimes n : v_h \otimes n \right),
+#   \\
+#   L_{c_h}(v_h) &= - \int_{\partial \Omega} u_D \cdot n \hat{u}_D \cdot
+#   v_h, \\
 #   b_h(v, q) &= - \int_K \nabla \cdot v q.
 # \end{align}
 # $$

python/demo/demo_pml.py

@@ -1,4 +1,4 @@
-# # Electromagnetic scattering from a wire with perfectly matched layer condition
+# # Electromagnetic scattering from a wire with perfectly matched layer condition # noqa
 #
 # Copyright (C) 2022 Michele Castriotta, Igor Baratta, Jørgen S. Dokken
 #
@@ -221,7 +221,8 @@ def generate_mesh_wire(
 # being hit normally by a TM-polarized electromagnetic wave.
 #
 # The formula are taken from:
-# Milton Kerker, "The Scattering of Light and Other Electromagnetic Radiation",
+# Milton Kerker, "The Scattering of Light and Other Electromagnetic
+# Radiation",
 # Chapter 6, Elsevier, 1969.
 #
 # ## Implementation
@@ -489,11 +490,12 @@ def pml_coordinates(x: ufl.indexed.Indexed, alpha: float, k0: complex, l_dom: fl
 #
 # $$
 # \begin{align}
-# \text{PML}_\text{corners} \rightarrow \mathbf{r}^\prime &= (x^\prime, y^\prime) \\
+# \text{PML}_\text{corners} \rightarrow \mathbf{r}^\prime &= (x^\prime,
+#   y^\prime) \\
 # \text{PML}_\text{rectangles along x} \rightarrow
-#                                       \mathbf{r}^\prime &= (x^\prime, y) \\
+#   \mathbf{r}^\prime &= (x^\prime, y) \\
 # \text{PML}_\text{rectangles along y} \rightarrow
-#                                       \mathbf{r}^\prime &= (x, y^\prime).
+#   \mathbf{r}^\prime &= (x, y^\prime).
 # \end{align}
 # $$
 #
@@ -662,8 +664,8 @@ def create_eps_mu(
 # The final weak form in the PML region is:
 #
 # $$
-# \int_{\Omega_{pml}}\left[\boldsymbol{\mu}^{-1}_{pml} \nabla \times \mathbf{E}
-# \right]\cdot \nabla \times \bar{\mathbf{v}}-k_{0}^{2}
+# \int_{\Omega_{pml}}\left[\boldsymbol{\mu}^{-1}_{pml} \nabla \times
+# \mathbf{E} \right]\cdot \nabla \times \bar{\mathbf{v}}-k_{0}^{2}
 # \left[\boldsymbol{\varepsilon}_{pml} \mathbf{E} \right]\cdot
 # \bar{\mathbf{v}}~ d x=0,
 # $$

python/demo/demo_poisson.py

@@ -43,7 +43,8 @@
 # $$
 # \begin{align}
 #   a(u, v) &:= \int_{\Omega} \nabla u \cdot \nabla v \, {\rm d} x, \\
-#   L(v)    &:= \int_{\Omega} f v \, {\rm d} x + \int_{\Gamma_{N}} g v \, {\rm d} s.
+#   L(v)    &:= \int_{\Omega} f v \, {\rm d} x + \int_{\Gamma_{N}} g v \,
+#               {\rm d} s.
 # \end{align}
 # $$
 #

python/demo/demo_poisson_matrix_free.py

@@ -107,9 +107,10 @@
 #
 # Next, we locate the mesh facets that lie on the domain boundary
 # $\partial\Omega$. We do this by first calling
-# {py:func}`create_connectivity <dolfinx.mesh.topology.create_connectivity>`
-# and then retrieving all facets on the boundary using
-# {py:func}`exterior_facet_indices <dolfinx.mesh.exterior_facet_indices>`.
+# {py:func}`create_connectivity
+# <dolfinx.mesh.topology.create_connectivity>`  and then retrieving all
+# facets on the boundary using {py:func}`exterior_facet_indices
+# <dolfinx.mesh.exterior_facet_indices>`.
 
 tdim = mesh.topology.dim
 mesh.topology.create_connectivity(tdim - 1, tdim)
@@ -156,10 +157,11 @@
 
 # ### Matrix-free conjugate gradient solver
 #
-# The right hand side vector $b - A x_{\rm bc}$ is the assembly of the linear
-# form $L$ where the essential Dirichlet boundary conditions are implemented
-# using lifting. Since we want to avoid assembling the matrix `A`, we compute
-# the necessary matrix-vector product using the linear form `M` explicitly.
+# The right hand side vector $b - A x_{\rm bc}$ is the assembly of the
+# linear form $L$ where the essential Dirichlet boundary conditions are
+# implemented using lifting. Since we want to avoid assembling the matrix
+# `A`, we compute the necessary matrix-vector product using the linear form
+# `M` explicitly.
 
 # Apply lifting: b <- b - A * x_bc
 b = fem.assemble_vector(L_fem)

python/demo/demo_pyamg.py

@@ -58,7 +58,8 @@ def poisson_problem(dtype: npt.DTypeLike, solver_type: str):
 
     Args:
         dtype: Scalar type to use.
-        solver_type: pyamg solver type, either "ruge_stuben" or "smoothed_aggregation"
+        solver_type: pyamg solver type, either "ruge_stuben" or
+            "smoothed_aggregation"
     """
 
     real_type = np.real(dtype(0)).dtype
@@ -191,7 +192,8 @@ def elasticity_problem(dtype):
     λ = E * ν / ((1.0 + ν) * (1.0 - 2.0 * ν))
 
     def σ(v):
-        """Return an expression for the stress σ given a displacement field"""
+        """Return an expression for the stress σ given a displacement
+        field"""
         return 2.0 * μ * ufl.sym(ufl.grad(v)) + λ * ufl.tr(ufl.sym(ufl.grad(v))) * ufl.Identity(
             len(v)
         )