diff --git a/hog/forms_facets.py b/hog/forms_facets.py
index daff842ae003bd86ca62323cbeb3d6c10665fab6..47292dad5766e463b81536529de5304aaa0c67de 100644
--- a/hog/forms_facets.py
+++ b/hog/forms_facets.py
@@ -179,7 +179,9 @@ def stokes_p0_stabilization(
r"""
Interface integrals for the stabilization term
- \gamma |\meas{e}| ( [[p_h]]_e, [[q_h]]_e )_e
+ .. math::
+
+ \gamma |\text{meas}(e)| ( [[p_h]]_e, [[q_h]]_e )_e
for the P1-P0 Stokes discretization as found in
@@ -189,13 +191,18 @@ def stokes_p0_stabilization(
Interface integrals are performed from both sides. interface_type has to be
- 'inner' coupling of element unknowns with themselves
+ 'inner' coupling of element unknowns with themselves
+
+ .. math::
+
+ \int_e \gamma \cdot \text{meas}(e) p q
- \int_e \gamma \meas{e} p q
+ 'outer' coupling of element unknowns with those opposite of the interface
- 'outer' coupling of element unknowns with those opposite of the interface
+ .. math::
+
+ - \int_e \gamma \cdot \text{meas}(e) p q
- - \int_e \gamma \meas{e} p q
"""
if not isinstance(blending, IdentityMap):
@@ -306,24 +313,29 @@ def diffusion_sip_facet(
All "interface types" are handled by this function. The type has to be specified by the interface_type argument:
Possible types:
- 'inner': coupling of element unknowns with themselves (both, v and w restricted to E1)
+ 'inner': coupling of element unknowns with themselves (both, v and w restricted to E1)
+
+ .. math::
- - 0.5 * \int_e J_v^{-\top} \nabla v \cdot n w
- - 0.5 * \int_e J_w^{-\top} \nabla w \cdot n v
- + \frac{\sigma}{\meas(e)^\beta} \int_e v w
+ - 0.5 \cdot \int_e J_v^{-\top} \nabla v \cdot n w
+ - 0.5 \cdot \int_e J_w^{-\top} \nabla w \cdot n v
+ + \frac{\sigma}{\text{meas}(e)^\beta} \int_e v w
- 'outer': coupling of element unknowns with those opposite of the interface (v restricted to E1, w restricted to
- E2)
+ 'outer': coupling of element unknowns with those opposite of the interface (v restricted to E1, w restricted to E2)
- + 0.5 * \int_e J_v^{-\top} \nabla v \cdot n w
- - 0.5 * \int_e J_w^{-\top} \nabla w \cdot n v
- - \frac{\sigma}{\meas(e)^\beta} \int_e v w
+ .. math::
- 'dirichlet': interface is at Dirichlet boundary (both, v and w restricted to E1)
+ + 0.5 \cdot \int_e J_v^{-\top} \nabla v \cdot n w
+ - 0.5 \cdot \int_e J_w^{-\top} \nabla w \cdot n v
+ - \frac{\sigma}{\text{meas}(e)^\beta} \int_e v w
- - \int_e J_v^{-\top} \nabla v \cdot n w
- - \int_e J_w^{-\top} \nabla w \cdot n v
- + 2 \frac{\sigma}{\meas(e)^\beta} \int_e v w
+ 'dirichlet': interface is at Dirichlet boundary (both, v and w restricted to E1)
+
+ .. math::
+
+ - \int_e J_v^{-\top} \nabla v \cdot n w
+ - \int_e J_w^{-\top} \nabla w \cdot n v
+ + 2 \frac{\sigma}{\text{meas}(e)^\beta} \int_e v w
Implementation:
@@ -336,12 +348,13 @@ def diffusion_sip_facet(
The integrals are evaluated over e in reference space of e (S_e).
v, w, \nabla v and \nabla w are evaluated on the _element_ reference space. This involves trafos
- T_e^{E1}: S_e \rightarrow S_E1, and T_e^{E2}: S_e \rightarrow S_E2. So v above is actually v( T_e^{E1}( x ) ).
+ :math:`T_e^{E1}: S_e \rightarrow S_{E1}`, and :math:`T_e^{E2}: S_e \rightarrow S_{E2}`. So v above is actually
+ :math:`v( T_e^{E1}( x ) )`.
The Jacobians are formed from the transformation from S_E1 to S and S_E2 to S.
Parameters: the first set of affine vertices belongs to the interior element, the second set to the neighboring
- element. An additional set of affine vertices corresponds to the interface vertices. Overall, e.g. in 2D we have
+ element. An additional set of affine vertices corresponds to the interface vertices. Overall, e.g. in 2D we have::
p_0, p_1, p_2 for the inner triangular element
p_3, p_4, p_5 for the outer triangular element
@@ -350,7 +363,7 @@ def diffusion_sip_facet(
p_9 outer triangle vertex that is not on the interface
p_10 for the normal from E1 to E2
- In 3D we get
+ In 3D we get::
p_0, p_1, p_2, p_3 for the inner tet element
p_4, p_5, p_6, p_7 for the outer tet element
@@ -512,7 +525,9 @@ def diffusion_sip_rhs_dirichlet(
This evaluates
- - \int_e ( \nabla v \cdot n + \frac{\sigma}{\meas(e)^\beta} v ) g_D
+ .. math::
+
+ - \int_e \left( \nabla v \cdot n + \frac{\sigma}{\text{meas}(e)^\beta} v \right) g_D
where g_D is an external function that evaluates to the Dirichlet boundary condition.
@@ -532,14 +547,14 @@ def diffusion_sip_rhs_dirichlet(
The Jacobians are formed from the transformation from S_E1 to S and S_E2 to S.
Parameters: the first set of affine vertices belongs to the interior element, the second set to the neighboring
- element. An additional set of affine vertices corresponds to the interface vertices. Overall, e.g. in 2D we have
+ element. An additional set of affine vertices corresponds to the interface vertices. Overall, e.g. in 2D we have::
p_0, p_1, p_2 for the inner triangular element
p_6, p_7 for the interface
p_8 inner triangle vertex that is not on the interface
p_10 for the normal from E1 to E2
- In 3D we get
+ In 3D we get::
p_0, p_1, p_2, p_3 for the inner tet element
p_8, p_9, p_10 for the interface
diff --git a/hog/function_space.py b/hog/function_space.py
index af4088369061277242670f6742b05fd230a9cdc1..1b576c275b454e12d2d561b9f70c04ddb0f660a4 100644
--- a/hog/function_space.py
+++ b/hog/function_space.py
@@ -80,14 +80,14 @@ class FunctionSpace(ABC):
) -> List[sp.MatrixBase]:
"""Returns a list containing the gradients of the shape functions on the element.
- Particularly, for each (vector- or scalar-valued) shape function N = (N_1, ..., N_n) returns the _transposed_
- Jacobian
+ Particularly, for each (vector- or scalar-valued) shape function N = (N_1, ..., N_n) returns the *transposed*
+ Jacobian::
- ⎡∂N₁/∂x₁ ··· ∂Nₙ/∂x₁⎤
- ⎢ · · ⎥
- grad(N) = J(N)ᵀ = ⎢ · · ⎥
- ⎢ · · ⎥
- ⎣∂N₁/∂xₖ ··· ∂Nₙ/∂xₖ ⎦
+ ⎡∂N₁/∂x₁ ··· ∂Nₙ/∂x₁⎤
+ ⎢ · · ⎥
+ grad(N) = J(N)ᵀ = ⎢ · · ⎥
+ ⎢ · · ⎥
+ ⎣∂N₁/∂xₖ ··· ∂Nₙ/∂xₖ⎦
i.e., the returned gradient is a column vector for scalar shape functions.
@@ -364,7 +364,7 @@ class TensorialVectorFunctionSpace(FunctionSpace):
shape functions of the given scalar function space.
For instance, a tensorial function space of dim d = 3 that is based on a P1 (Lagrangian) scalar function space with
- n shape functions N_1, ..., N_n has the shape functions
+ n shape functions N_1, ..., N_n has the shape functions::
(N_1, 0, 0),
...
@@ -377,7 +377,7 @@ class TensorialVectorFunctionSpace(FunctionSpace):
(0, 0, N_n).
This class also enables specifying only a single component to get the shape functions, e.g. only for component 1
- (starting to count components at 0)
+ (starting to count components at 0)::
(0, N_1, 0),
...