Discretization by rational and quasi-rational functions of multi-dimensional elliptic problems in the whole space

Abstract

We propose a new spectral method for solving multi-dimensional second order elliptic equations with varying coefficients in the whole space. This method employs an orthogonal family of quasi-rational functions recently discovered by Arar and Boulmezaoud. After proving an error estimate, we present some computational tests which demonstrate the efficiency of the method and the significance of its developmental potential.

A simple formula of the magnetic potential and of the stray field energy induced by a given magnetization

“…Let 𝜋 N be the orthogonal projector on H N with respect to the scalar product associated to the norm |.| W 1 0 (R 3 ) . The following result is due to Boulmezaoud et al [35]:…”

“…which is equivalent to the norm ||.|| W 1 0 (R 3 ) . The following lemma will play a prominent role in the sequel (see Arar & Boulmezaoud [34] and Boulmezaoud et al [35]):…”

“…In the three-dimensional situation (the only one that interests us here), special functions we need here are defined as follows (see Arar & Boulmezaoud [34] and Boulmezaoud et al [35]): for any 𝛼 ∈ Λ…”

“…This is a general property as it is announced in the following proposition which summarizes some useful properties of the functions (W 𝛼 ) 𝛼∈Λ . We refer to Arar and Boulmezaoud [34] and Boulmezaoud et al [35] for the proofs.…”

“…where (W 𝛼 ) 𝛼 designates a special family of multidimensional functions introduced in Arar and Boulmezaoud [34] and in Boulmezaoud et al [35] (these functions will be presented along with their properties in Section 2). The index sets Λ k , k ⩾ 0, are defined hereafter by (16).…”