THE DISCONTINUOUS GALERKIN APPROXIMATION OF THE GRAD-DIV AND CURL-CURL OPERATORS IN FIRST-ORDER FORM IS INVOLUTION-PRESERVING AND SPECTRALLY CORRECT

The discontinuous Galerkin approximation of the grad-div and curl-curl problems formulated in conservative first-order form is investigated. It is shown that the approximation is spectrally correct, thereby confirming numerical observations made by various authors in the literature. This result hinges on the existence of discrete involutions which are formulated as discrete orthogonality properties. The involutions are crucial to establish discrete versions of weak Poincar\'e-Steklov inequalities that hold true at the continuous level.