Discrete Hodge-Operators: an Algebraic Perspective

Discrete differential forms should be used to deal with the discretization of boundary value problems that can be stated in the calculus of differential forms. This approach preserves the topological features of the equations. Yet, the discrete counterparts of the metric- dependent constitutive laws remain elusive. I introduce a few purely algebraic constraints that matrices associated with discrete material laws have to satisfy. It turns out that most finite element and finite volume schemes comply with these requirements. Thus convergence analysis can be conducted in a unified setting. This discloses basic sufficient conditions that discrete material laws have to meet in order to ensure convergence in the relevant energy norms.

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