Rigidity in Finite-Element Matrices: Sufficient Conditions for the Rigidity of Structures and Substructures

We present an algebraic theory of rigidity for finite-element matrices. The theory provides a formal algebraic definition of finite-element matrices; notions of rigidity of finite-element matrices and of mutual rigidity between two such matrices; and sufficient conditions for rigidity and mutual rigidity. We also present a novel sparsification technique, called fretsaw extension, for finite-element matrices. We show that this sparsification technique generates matrices that are mutually-rigid with the original matrix. We also show that one particular construction algorithm for fretsaw extensions generates matrices that can be factored with essentially no fill. This algorithm can be used to construct preconditioners for finite-element matrices. Both our theory and our algorithms are applicable to a wide-range of finiteelement matrices, including matrices arising from finite-element discretizations of both scalar and vector partial differential equations (e.g., electrostatics and linear elasticity). Both the theory and the algorithms are purely algebraic-combinatorial. They manipulate only the element matrices and are oblivious to the geometry, the material properties, and the discretization details of the underlying continuous problem.