This part of the book is devoted to the finite element approximation to solutions of A.2.11). The principal aim of the present Chapter 2 is the robust exponential convergence result Theorem 2.4.8, which is illustrated by numerical examples in Section 2.5. Essential for this robust exponential convergence result are detailed regularity assertions for the solution. For the convenience of the reader, the present chapter collects from Parts II, III the regularity results that are required for the proof of Theorem 2.4.8. The proofs of both the approximation result and the regularity assertions are very technical and therefore not included in this chapter. In order to motivate the two-dimensional results of this chapter, we present the analogous results in the one-dimensional setting in Section 2.2. Technically, this setting is considerably simpler than the two-dimensional case, yet it exhibits many features that are relevant for the two-dimensional case. We conclude this chapter with a discussion of a low-order method in Section 2.6, since the regularity assertions of Section 2.3 can also be employed to prove robust convergence of the /i-FEM on Shishkin meshes.

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