Analysis I by Terence Tao

By Terence Tao

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1 are of the same order as the terms left out of the truncated asymptotic expansion. 9 It is possible to derive an error estimate under the presence of a cut-off function using the following result. 18). 4 Conclusions In general, it is not easy to design numerical methods schemes for singular perturbed PDEs. The layers that characterize such problems cause numerical instabilities that are hard to tame. Understanding such layers is then important, and there is no easier way to do so than considering one-dimensional cases.

1. 1 The smoothness of the domain is important here. In particular for a polygon, the above transformation is not a diffeomorphism.  / D . X 0 Y 00 ; Y 0 C X 00 /; and thus  r. 6), and < 0 , we obtain that det r . 5). Inverting the above matrix, we have that rx where J.  /. 7) 52 4 Two-Dimensional Reaction-Diffusion Equations Finally, the change of coordinates yields @f @f @ @f @ D C ; @˛ @ @˛ @ @˛ for ˛ D 1; 2; for an arbitrary function f . x/ C "2 u. x/ C . " where is a smooth cut-off function identically one if 0 Ä zero if 2 0 =3.

For higher dimensional problems, the introduction of the cut-off function is “mandatory,” as the boundary correctors are defined only in a neighborhood of the boundary, and not in the whole domain as here. See Chap. 4. 0; 1/, jrj Ä e 1=" is exponentially small with respect to 1=". 3 A More General Singular Perturbed Second Order ODE The asymptotic expansion of Sect. 2 is remarkably simple, but that does not represent the general case. 16) We assume that " > 0, that ˇ, , and f are smooth functions, and that both Cˇ=2 > 0 and ˇ > 0 are positive.

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