\(\newcommand{\opdiv}{\operatorname{div}}\) \(\newcommand{\optrn}{\operatorname{tr}_n}\)

1. The Function Space \(H(\operatorname{div})\)

A vector valued function space with well-defined divergence:

\[ H(\opdiv) = \{ \sigma \in [L_2(\Omega)]^2 : \opdiv \sigma \in L_2 \} \]

Has continuous normal trace operator:

\[\begin{eqnarray*} \operatorname{tr} : H(\opdiv) & \rightarrow & H^{-1/2}(\partial \Omega) \\ \sigma & \rightarrow & \sigma \cdot n \quad \text{for smooth $\sigma$} \end{eqnarray*}\]

Functions from \(H(\opdiv)\) have single-valued normal components across sub-domain boundaries.

Brezzi-Douglas-Marini BDM finite elements:

\[ {\mathcal BDM}_k = \{ \sigma_h \in [P_k({\mathcal T})]^2 : [\sigma_h \cdot n] = 0 \} \]

very similar: Raviart-Thomas RT elements

1.1. Piola transformation:

mapping of domains: $\( x = \Phi (\hat x) \)$

define Jacobian and determinant:

\[ F = \Phi^\prime, \; J = \det(F) \]

Push-forward of vector fields by Piola transformation

\[ {\mathcal P} \sigma (x) := \frac{1}{J} F \hat \sigma(\hat x) \]

from ngsolve import *
from ngsolve.webgui import Draw
from netgen.occ import *
import ipywidgets as widgets
from ipywidgets import interact, fixed

geo = MoveTo(0,-0.1).Rectangle(1,0.2).Face()
mesh = Mesh(OCCGeometry(geo, dim=2).GenerateMesh(maxh=0.05))

fes = HDiv(mesh, order=2)
# fes = VectorH1(mesh, order=2)

sigma = GridFunction(fes)
sigma.Set ( (2-100*y*y, 0))

disp = GridFunction(VectorH1(mesh, order=2))
scene = Draw (sigma, vectors={"grid_size":30}) 

def myfunc(bend, thick): 
    disp.Set((-2*bend*x*y-0.5*y*y*(thick-1), bend*x*x + x*y*(thick-1)))
    mesh.SetDeformation(disp)
    scene.Redraw()
    mesh.UnsetDeformation()
    
    
params = widgets.interactive(myfunc,
        bend=widgets.FloatSlider(value=0, min=-0.2, max=0.2, step=0.001, 
                                                    description='bend:', disabled=False, 
                                                    continuous_update=True, orientation='horizontal',
                                                    readout=True, readout_format='.3f'),
        thick=widgets.FloatSlider(value=1, min=0.01, max=2, step=0.001, 
                                                    description='thick:', disabled=False, 
                                                    continuous_update=True, orientation='horizontal',
                                                    readout=True, readout_format='.3f'),)


display (params)

2. Goal:

• use \(H(\operatorname{div})\) function space for approximation of flow velocities

• perfect for divergence free constraints

• discretization of viscosity term ?

  • HDG methods

  • mixed stress methods