Isomorphic Structures and Operator Analysis in Mimetic Discretizations

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Abstract

This study presents a comprehensive examination of the structural and operatorial foundations within mimetic discretizations, with a focus on bridging the gap between discrete and continuous function spaces. By scrutinizing the mimetic gradient and divergence operators-central to the discretization of the NAVIER-STOKES equations-we study their kernel and image spaces, establishing their isomorphisms through rigorous mathematical proofs. Our methodology leverages discrete scalar and vector function spaces, delineated by grid spacing, to define linear mappings that unveil the subspace relationships and quotient space structures integral to understanding these operators' roles in computational fluid dynamics. Central to our findings is the application of the first isomorphism theorem, which facilitates a deeper insight into how mimetic discretizations reflect the continuous properties of differential operators within a discrete framework. This allows for an exploration into the algebraic and topological implications of such discretizations, notably in the context of the NAVIER-STOKES equations. Furthermore, we extend our investigation to encompass subalgebras, ideals, their quotients, and the formulation of short exact sequences that mirror the continuous interplay between gradient, divergence, and LAPLACIAN operators. Significant advances include the application of the first isomorphism theorem which confirms that our mimetic discretizations preserve key properties of differential operators, thus enhancing the accuracy and reliability of computational models. Additionally, our research introduces practical extensions into subalgebras and complex operator sequences, laying groundwork for future developments in numerical methods aimed at improving the precision of engineering simulations.