DeRham sequences¶
here without boundary conditions
Pullbacks¶
In the case where the physical domain is the image of a logical domain by a smooth mapping (at least ), we have the following parallel diagrams
Where the mappings and are called pullbacks and are given by
where is the jacobian matrix of the mapping .
Note
The pullbacks and are isomorphisms between the corresponding spaces.
Discrete Spaces¶
Let us suppose that we have a sequence of finite subspaces for each of the spaces involved in the DeRham sequence. The discrete DeRham sequence stands for the following commutative diagram between continuous and discrete spaces
When using a Finite Elements methods, we often deal with a reference element, and thus we need also to apply the pullbacks on the discrete spaces. In fact, we have again the following parallel diagram
Since, the pullbacks are isomorphisms in the previous diagram, we can define a one-to-one correspondance
We have then, the following results
Discrete DeRham sequence for B-Splines¶
Buffa et al [BSV09] show the construction of a discrete DeRham sequence using B-Splines, (here without boundary conditions)
1d case¶
- DeRham sequence is reduced to
- The recursion formula for derivative writes
- we have which is a change of basis as a diagonal matrix
- Now if , with and expansion , we have
- If we introduce the B-Splines coefficients vector (and for the derivative), we have
where is the incidence matrix (of entries and )
Discrete derivatives:
2d case¶
In 2d, the are two De-Rham complexes:
and
Let be the identity matrix, we have
Discrete derivatives:
3d case¶
Discrete derivatives:
Note
From now on, we will denote the discrete derivative by for the one going from to .
Algebraic identities¶
Let us consider the discretization of the exterior derivative
multiplying by a test function and integrating over the whole computation domain, we get
let , and be the vector representation of , and . We get
where
On the other hand, using the coderivative, we get
Let us now introduce the following matrix
hence,
Therefor, we have the following important result
Proposition:
References
[BSV09] | A. Buffa, G. Sangalli, and R. Vazquez. Isogeometric analysis in electromagnetics: b-splines approximation. Comput. Methods Appl. Mech. Engrg, 199:1143–1152, 2009. |