Exponential node clustering at singularities for rational approximation, quadrature, and PDEs

Reciprocal-Log Approximation and Planar PDE Solvers

“…The poles of our approximations g(z) lie exponentially close to the origin in the z-plane, at least when the formula (2.3) is used. Exponential clustering of poles is familiar in rational approximation theory [20]. Another phenomenon appears in Figure 2 [1]; the numbers obtained without stabilization are shown by the small dots.)…”

“…With both best and near-best approximations, one encounters the startling property that the oscillations typically cluster doubleexponentially near the singularity, readily coming as close (in theory) as a distance of, say, 10 - 1000 . This is in contrast to the well-known case of rational approximation, where the clustering is just exponential [20]. Section 3 generalizes the discussion to domains in the complex z-plane with m \geq 1 singularities, typically at corners.…”

Rectangular eigenvalue problems

“…Specifically, following eq. (3.2) of [16], a formula that is justified in [33], Fig. 1 Maximum error in the first six computed eigenvalues of the singular Schrödinger problem (15).…”

Lightning Stokes solver

“…For the simple poles, we choose points β kn with tapered exponential clustering near the corners w k , following the formula introduced in [13] and analyzed in [40], (4.3)…”

“…In this paper we propose a numerical method to solve (1.1)-(1.3) by generalizing the recently introduced "lightning solvers" for the 2D Laplace and Helmholtz equations [12,13,40]. The main advantage of this class of methods is that they can handle domains with corners without requiring any detailed analysis, and still achieve high accuracy, with root-exponential convergence as a function of the number of degrees of freedom.…”