Superconvergence of discontinuous Galerkin methods for two-dimensional hyperbolic equations

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© 2015 Society for Industrial and Applied Mathematics. This paper is concerned with superconvergence properties of discontinuous Galerkin (DG) methods for two-dimensional linear hyperbolic conservation laws over rectangular meshes when upwind fluxes are used. We prove, under some suitable initial and boundary discretizations, the (2k + 1)th order superconvergence rate of the DG approximation at the downwind points and for the cell averages, when piecewise tensor-product polynomials of degree k are used. Moreover, we prove that the gradient of the DG solution is superconvergent with a rate of (k + 1)th order at all interior left Radau points; and the function value approximation is superconvergent at all right Radau points with a rate of (k + 2)th order. Numerical experiments indicate that the aforementioned superconvergence rates are sharp.

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SIAM Journal on Numerical Analysis