Recovery of normal derivatives from the piecewise L2 projection

In this paper, we propose a novel approach to recover normal derivatives for a smooth function based on its piecewise L2 projection. For each polynomial (of degree up to 4) projection, another polynomial of same degree (at least degree 1) is constructed over a sub-domain centered at the interface separating two polynomials, its normal derivative at interface is taken to be the recovered normal derivative. The pointwise accuracy of the recovery is shown to be of order 2m+12+2. From such a recovery algorithm we obtain a set of numerical flux formulae for solution derivatives. Following the direct discontinuous Galerkin (DDG) method introduced by Liu and Yan [H. Liu, J. Yan, The direct discontinuous Galerkin (DDG) method for diffusion with interface corrections, Commun. Comput. Phys. 8 (3) (2010) 541–564] for parabolic equations, we apply these flux formulae to some elliptic problems using polynomial elements of degree up to 4. Some adaptation of these numerical fluxes is adopted for even high order elements. Both one and two-dimensional numerical results are provided to demonstrate the good qualities of the recovery algorithm when combined with the DDG methods.