Towards a Variational Complex for the Finite Element Method

Can we use symbolic algebra to study numerical methods? • Can you design a numerical method, automatically, to inherit a variational principle and selected conservation laws? • Can one obtain symmetries and hence conservation laws, automatically, of variational numerical methods?

finite dimensional function spaces wrt some triangulation to the variational complex The preimages can be constructed.
Recall the smooth Euler Lagrange operator • Step 2: mod out by the boundary terms, that is, total derivatives.
The simplest FEM example using the mindset/technology/terminology of formal variational processes: The triangulation of R has vertices x n and edges e n = (x n , x n+1 ).The data is The projection is The discrete Lagrangian is Π(u) 2 x − 2Π(u) and hence The blue term telescopes and hence is set to zero: this term is a boundary term.
Using the identity Since each δu n is independent, the coefficient must be zero (at least in the function space 2 ), and this then is the discrete Euler Lagrange equation.Integrating twice gives Not all choices of Finite Element are suited to variational methods Using piecewise constant functions to approximate functions leads to poor results.Instead, use the zeroth moments for u on (x n , x n+1 ) and (x n+1 , x n+2 ) to create a piecewise linear approximation for u on (x n , x n+2 ) with the same 2 moments: for an approximation on the partition This fixed projection fits into a coherent scheme.
In For the face in the diagram, the boundary is In general, we write, The map δ is called the coboundary operator.
If you are using an interpolation scheme, all the data lie on the vertices.The set of faces is then a set F of ordered triples of indices and the set of edges E is a set of ordered pairs of indices.
The ordering gives the orientation. (ikj) A telescoping/coboundary sum looks like that is, cyclic sums on edges.
The projection of X L[u] dx is where |f i | = 1 if f i has the anti-clockwise orientation, and |f i | = −1 for the reverse orientation.
If Π(L) is a coboundary, the sums over the internal edges will cancel: (1) will depend only on the boundary data.In fact we have a discrete Stokes' theorem, The simplicial theory is attractive.It allows us to use results and intuition from classical work on triangulations.It generalizes to n dimensions.
From FE forms to simplicial cochains Let the top, i.e. n dimensional simplices be denoted by τ .Given a piecewise defined n-form on the τ , a simplicial n-cochain is achieved by integrating the form on the τ .This map is the de Rham map.We will denote it by :

Theorem
The FE variational complex, shown here for a three dimensional base space, is locally exact: the kernel of one map equals the preimage of the previous.The preimage can be constructed.
F * is the algebra generated by the W h , Q h . . .with unevaluated degrees of freedom 2. Develop a computationally useful notion of the EL equations in terms of the incidence matrix of the mesh.
3. Derive formulae for discrete Noether's theorem relative to a given mesh.
where D as above but now we have smooth functions of x, u, u x ,. . .with derivatives acting totally E ≡ Euler Lagrange operator Theorem: Locally, the kernel of one map equals the image of the previous in all these sequences.
is the analogue of the vertical exterior derivative, modulo boundary termsF ** is the algebra of vertical forms tensored with the space of n-dimensional simplicial co-chains Proofs and details appear in the preprint "Towards a variational complex for the Finite Element Method", to appear in a Centre de Recherches Mathematiques Proceedings volume for the workshop "Group Theory and Numerical Analysis", May 2003.Available from the URL http://www.kent.ac.uk/ims/personal/elm2Incorporate the Poincaré dual mesh model of the Hodge star operator.