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In this paper, a new DG method was designed to solve the model problem of the one-dimensional singularly-perturbed convection-diffusion equation. With some special chosen numerical traces, the existence and uniqueness of the DG solution is provided. The superconvergent points inside each element are observed.Particularly, the 2p + 1-order superconvergence and even uniform superconvergence under layer-adapted mesh are observed numerically.

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