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Indeed, it occurs whenever the target region has areas that are relatively long and thin. We informally call such regions elongated. One interpretation of crowding is that whereas boundary data for an elliptic partial differential equation influence the solution throughout the region, this influence may be exponentially weak in areas separated from the point by an elongation. 4 (see also [Lau94, PS91, FPS99]). 3 (see also [How94]). This page intentionally left blank 3 Numerical methods Methods for the numerical computation of Schwarz–Christoffel maps have been refined for 40 years, and improvements continue.
To locate wn , we note that the lines containing the sides that adjoin wn are not parallel, because of the conditions on αn . Since the angles at w1 and wn−1 are known, we can locate wn at the unique intersection of two nonparallel lines. 1. The conditions on αn are not restrictive in practice, as we are free to number the vertices so that a convenient vertex is the nth. The scaling, rotation, and wn wn−1 w1 w3 w2 ... 1. Location of the last vertex by intersection. 1). 2). 2) are meaningless. We can replace them by the complex condition z J +1 z J −1 z2 f (ζ ) dζ = f (ζ ) dζ w J +1 − w J −1 .
The SC Toolbox is a descendent of SCPACK, a FORTRAN 77 package written by Trefethen [Tre80, Tre89]. SCPACK was responsible for first making numerical SC mapping widely available to nonexperts and has been in use for 20 years. Only maps from the disk, using the side-length formulation of the parameter problem, are directly supported; routines are included for computing the forward and inverse maps. org/conformal. 9). As in SCPACK, solving the parameter problem and computing maps in either direction are supported.
A-stable block implicit one-step methods