Norm-preserving discretization of integral equations for elliptic PDEs with internal layers I: the one-dimensional case

We investigate the behavior of integral formulations of variable coefficient elliptic partial differential equations (PDEs) in the presence of steep internal layers. In one dimension, the equations that arise can be solved analytically and the condition numbers estimated in various L_p norms. We show that high-order accurate Nyström discretization leads to well-conditioned finite dimensional linear systems if and only if the discretization is both norm-preserving in a correctly chosen L_p space and adaptively refined in the internal layer.

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