Multivariate modified Fourier series and application to boundary value problems

Abstract

In this paper we analyse the approximation-theoretic properties of modified Fourier series in Cartesian product domains with coefficients from both full and hyperbolic cross index sets. We show that the number of expansion coefficients may be reduced significantly whilst retaining comparable error estimates. In doing so we extend the univariate results of Iserles, Nørsett and S. Olver. We then demonstrate that these series can be used in the spectral-Galerkin approximation of second order Neumann boundary value problems, which offers some advantages over standard Chebyshev or Legendre polynomial discretizations.

Convergence acceleration of modified Fourier series in one or more dimensions

“…Much like the Fourier case, this is due to "jumps" in certain derivatives of the function at the endpoints x = ±1 (in the univariate case) [28]. In the multivariate setting, similar analogues hold, although the jump conditions (otherwise referred to as derivative conditions) are more complicated to express [1,16].…”

“…In the univariate case, the Fourier sine function is replaced by sin(n − 1 2 )πx, yielding the basis {cos nπx, n ∈ N} ∪ {sin(n − 1 2 )πx, n ∈ N + }. The multivariate extension is obtained by Cartesian products.…”

“…The advantage of this basis is that the modified Fourier expansion of a sufficiently smooth function f converges uniformly onΩ. In particular, there is no Gibbs phenomenon near the boundary [1,17,28].…”

“…If N is a truncation parameter, the uniform error is O N −1 onΩ and O N −2 inside compact subsets of Ω [1,17,28]. Much like the Fourier case, this is due to "jumps" in certain derivatives of the function at the endpoints x = ±1 (in the univariate case) [28].…”

“…However, it happens that this figure can be significantly reduced to O N (log N ) d−1 by using a so-called hyperbolic cross index set [3,32]. The use of such an index set does not deteriorate the convergence rate, aside from possibly a logarithmic factor [32] (for application of this index set to modified Fourier expansions see [1,16]). In the final part of this paper we demonstrate how to incorporate such an index set into Eckhoff's method.…”