Applications of functional analysis in mathematical physics

Polynomial approximation of functions in Sobolev spaces

“…Morrey [16] (and implicitly in Sobolev [18]) for the case of P being all polynomials of degree at most r, A being all multi-indices of length r + 1, || • || being the norm on W™, and |-1 being the norm on L . In the second Bramble-Hilbert paper [5], (1.1) is derived for certain classes P that range from the polynomials of degree at most r to the polynomials that are of degree at most r in each variable separately.…”

“…The work of Ciarlet and Wagschal on multipoint Taylor formulas [7] is another approach to giving constructive proofs of approximation results needed for finite element analysis and their results played an important role in the evolution of this paper. The basic representation given in Section 3, which we call a Sobolev representation, is quite similar to one used by Sobolev [18] in proving imbedding theorems (see Remark 3.5). However, it appears to be different from the one used in [18] for which, in particular, it is not clear that the commutativity property mentioned above is valid.…”

“…The basic representation given in Section 3, which we call a Sobolev representation, is quite similar to one used by Sobolev [18] in proving imbedding theorems (see Remark 3.5). However, it appears to be different from the one used in [18] for which, in particular, it is not clear that the commutativity property mentioned above is valid. A more recent treatment of related representations, as well as some discussion of their applications in other work, can be found in an article by Burenkov [6].…”

Extremum principles in electromagnetic systems

“…Moreover, by standard trace embedding theorems n x E and n x H both belong to L2(Of2) (see [15], Chapter IX), so that boundary conditions (3.3) are well-defined. In connection with problem ~1 we are able to prove the following THEOREM 3.1.…”

Error Analysis of Finite Element Methods With Triangles for Elliptic Boundary Value Problems

“…In the finite element context there are two immediate reasons. Firstly Sobolev [14] proves that the which minimizes I(v) over the subspace. For this finite dimensional subspace we use the notation S q , where q indicates the form of the approximation.…”