Polynomial approximation of divergence-free functions

Abstract

We study the best approximation of a divergence-free function by a divergence-free algebraic or trigonometric polynomial and we prove an optimal estimate. In a particular case we give also an optimal result for the polynomial approximation of a function and its divergence.

Analysis of spectral projectors in one-dimensional domains

“…Finally, we give in §5 an application to a simple fourth-order test problem. Note that other applications of these results can be found in [3,18].…”

“…Remark 2.3. As pointed out in [18], the operator A +P "o (Id-A ) defined in (2.7) for integral values of p , and that could be defined in a similar way for any real number p not in N + 1/4, has the same approximation properties as n N and, moreover, it preserves the traces of any element of HP(A). …”

Analysis of Spectral Projectors in One-Dimensional Domains

“…Moreover, it is proved in these two cases that no optimal bound is possible for H"(A)-norms with v > p . Indeed, the best estimate that can be achieved is \\<P-p0,N<p\l<CN2"~a\\<pV It is often necessary (see, e.g., [2,3,18]) to obtain optimal results in higher norms.…”

“…As noted in [18], the operator Pp N preserves the traces of any element of H^(A). This proves that in fact P N is independent of 5.…”

A Model for Two Coupled Turbulent Fluids Part II: Numerical Analysis of a Spectral Discretization

“…Indeed, the functions curl ψ iδ , where ψ iδ belongs to P Ki,Ni (Ω i ), satisfies the desired boundary conditions and approximates ψ i in H 2 (Ω i ), belongs to X iδ ∩V i , and provides a good approximation of u i . The case of dimension d = 3 is more complex; however, the right approximation properties have been proved in [22] for smooth functions and extended in [5] to arbitrary functions. So the general result reads as follows: for any…”