# How to attatch abstract domains to expressions

Suggested design.

# Cells are mapped internally in UFL to a default domain for compatibility:

_default_domains = {

triangle: Domain(triangle),

tetrahedron: Domain(

#...

}

def as_domain(D):

if isinstance(D, Cell):

return _default_domains[D]

return D

# Some kind of language for defining domains:

D = Domain(triangle) # A simple wrapper around cell can be the first step

E = Domain(triangle)

# Elements are defined on domains:

U = FiniteElement("CG", D, 1)

V = FiniteElement("CG", E, 1)

W = FiniteElement("CG", D, 2)

# Coefficients are defined on elements as before, but now has a domain associated implicitly:

f = Coefficient(U)

g = Coefficient(V)

h = Coefficient(W)

# Preprocessing a form will now let us recover the set of domains in use

a = f*g*h*dx

domains = a.compute_

assert domains == [D, E]

# There is a relation between the ordered list of coefficients and the ordered list of domains:

# w0 = f -> D = d0 # d0 = domain 0 just as w0 = coefficient 0

# w1 = g -> E = d1 # d1 = domain 1

# w2 = h -> D = d0

# This can be encoded as an integer mapping:

# { 0: 0, 1: 1, 2: 0 }

# These indices i:j map directly to the dofs w[i][] and cell c[j] in the upgraded ufc interface

# Coefficients in a form must be defined on the integration domain, this can be assumed and detected runtime

a = f*g*dx(D) # Assuming E subset of D such that g is defined in the entire integration domain

## Blueprint information

- Status:
- Complete

- Approver:
- Martin Sandve Alnæs

- Priority:
- Essential

- Drafter:
- Martin Sandve Alnæs

- Direction:
- Approved

- Assignee:
- Martin Sandve Alnæs

- Definition:
- Approved

- Series goal:
- None

- Implementation:
- Implemented

- Milestone target:
- None

- Started by
- Martin Sandve Alnæs

- Completed by
- Martin Sandve Alnæs

### Whiteboard

I'm calling this completed, although the description above does not match exactly what is done now. Make new blueprints with concrete suggestions for future changes and improvements related to this one.

### Work Items

Work items:

[martinal] Create Domain class with cell and name properties: DONE

[martinal] Create as_domain function and let finite elements take domain or cell: DONE

[martinal] Extract domains in form data: DONE

[martinal] Allow Domain as argument to Measure (preliminary design, map somehow to domain numbers for now): DONE

## Dependency tree

* Blueprints in grey have been implemented.