SUPERCONVERGENCE OF DG METHOD FOR ONE-DIMENSIONAL SINGULARLY PERTURBED PROBLEMS

The convergence and superconvergence properties of the discontinuous Galerkin (DG)method for a singularly perturbed model problem in one-dimensional setting are studied.By applying the DG method with appropriately chosen numerical traces,the existence anduniqueness of the DG solution,the optimal order Lerror bounds,and 2p+1-order super-convergence of the numerical traces are established.The numerical results indicate thatthe DG method does not produce any oscillation even under the uniform mesh.Numericalexperiments demonstrate that,under the uniform mesh,it seems impossible to obtain theuniform superconvergence of the numerical traces.Nevertheless,thanks to the implemen-tation of the so-called Shishkin-type mesh,the uniform 2p+1-order superconvergence isobserved numerically.