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In this paper, we uncover and study a new superconvergence property of a large class of finite element methods for one-dimensional convectiondiffusion problems. This class includes discontinuous Galerkin methods defined in terms of numerical traces, discontinuous Petrov-Galerkin methods and hybridized mixed methods. We prove that the so-called numerical traces of both variables superconverge at all the nodes of the mesh, provided that the traces are conservative, that is, provided they are single-valued. In particular, for a local discontinuous Galerkin method, we show that the superconvergence is order 2 p + 1 when polynomials of degree at most p are used. Extensive numerical results verifying our theoretical results are displayed.

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