Function Space concepts

Function Space concepts#

SymPDE provides two Python classes to describe scalar and vector function spaces, respectively. There is no notion of a discrete representation; for example, we do not need to mention that a space is a Brezzi-Douglas-Marini (BDM) space. In fact, these kind of function spaces can be seen as parametric types having as a basic type ScalarFunctionSpace or VectorFunctionSpace. SymPDE only needs to know if an element of the space can be indexed or not. A BDM space would then be identified by an annotation added by a third party library to uniquely define a function space at the discrete level.

BasicFunctionSpace#

This represents the base class of all our function spaces. It does not reflect a continuous space, but it is more a type at the formal/abstract level.

ScalarFunctionSpace#

>>> domain = Domain('Omega', dim=2)
>>> V1 = ScalarFunctionSpace('V1', domain)

VectorFunctionSpace#

>>> domain = Domain('Omega', dim=2)
>>> W1 = VectorFunctionSpace('W1', domain)

Typed Spaces#

>>> domain = Domain('Omega', dim=2)
>>> H1    = ScalarFunctionSpace('V0', domain, kind='H1')
>>> Hcurl = VectorFunctionSpace('V1', domain, kind='Hcurl')
>>> Hdiv  = VectorFunctionSpace('V2', domain, kind='Hdiv')
>>> L2    = ScalarFunctionSpace('V3', domain, kind='L2')
>>> assert(H1.kind    == H1Space)                
>>> assert(Hcurl.kind == HcurlSpace)
>>> assert(Hdiv.kind  == HdivSpace)
>>> assert(L2.kind    == L2Space)
>>> assert(V.kind     == UndefinedSpace)
>>> assert(W.kind     == UndefinedSpace)
>>> assert(H1.regularity    > L2.regularity)
>>> assert(H1.regularity    > Hcurl.regularity)
>>> assert(Hcurl.regularity > L2.regularity)

ProductSpace#

>>> domain = Domain('Omega', dim=2)
>>> V1 = ScalarFunctionSpace('V1', domain)
>>> V2 = ScalarFunctionSpace('V2', domain)
>>> W1 = VectorFunctionSpace('W1', domain)
>>> X = V1 * V2 * W1
>>> # or 
>>> X = ProductSpace(V1, V2, W1)

Derham#

Functions#

ScalarFunction#

A scalar function is necessary an element of the ScalarFunctionSpace.

IndexedVectorFunction#

VectorFunction#

A vector function is necessary an element of the VectorFunctionSpace. It is an extension of the IndexedVectorFunction, which means that one can access to its components.

Trace#

trace_0#

trace_1#

Utilities#

element_of and elements_of#

Projection (Experimental)#

Projector (Experimental)#

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