TWO OBSTRUCTIONS TO COERCIVITY: A UNIFIED CRITERION FOR KORN, MAXWELL, AND MIXED METHODS

A symmetric positive semidefinite form q fails to be coercive with respect to a seminorm p in exactly one of two ways: either p does not vanish on the radical of q, or the form induced on the quotient does not dominate the quotient seminorm. The two conditions are stated in §2 from the correspondence between symmetric PSD forms with prescribed radical and positive definite forms on the quotient. For forms of the type qA(v, v) = ∥Av∥ 2 W with p = pL = distV (·, ker A), the second condition is equivalent to the closed range property of A (§3). The paper applies that reduction to three problems of numerical analysis: nonconforming Korn, Maxwell with edge elements, and mixed finite element methods (§4–§6). In each case, the part of the analysis that follows from the criterion is separated from the part that is a design decision—the choice of penalization, of split, of formulation, or of pair of spaces (§7). The scope is stated in §7.4.