TWO OBSTRUCTIONS TO COERCIVITY: A RADICAL-QUOTIENT CONSTRUCTION FOR PSD BILINEAR FORMS - (with an extended appendix on families of discretizations)
Year: 2026
Type: preprint
Abstract: First Abstract. A symmetric positive semidefinite bilinear form is usually chosen a priori, and its radical is derived afterwards. This note formalizes the inverse procedure. The construction consists of five components of three types: two free data (L and g), one canonical construction (the quotient V /L), and two operations (assembly and level separation). No order is defined among the components. The correspondence between symmetric PSD forms with radical L and positive definite forms on V /L is a bijection. The failure of coercivity is separated into two exhaustive and mutually exclusive obstructions: the seminorm does not vanish on L, or the induced form on V /L does not dominate the quotient seminorm. The two obstructions correspond to the two decisions that the standard order does not distinguish: the choice of the free subspace (modeling) and the choice of the metric on the quotient (analysis). Second Abstract. The abstract construction is complemented by an appendix that extends the criterion to families of discretizations. Using the Hodge decomposition, the closed range theorem for closed operators, and uniform Poincar´e inequalities, the appendix provides sufficient conditions for the uniform coercivity of a family. The results are directly applicable to the stability analysis of discrete exterior calculus and the Hodge-Laplacian on Hilbert complexes.
Author: Mena-Zapata, Julian Armando
Language: English
Subfield: Computational Mechanics
Field: Engineering
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7225594310