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Abstract This work introduces a novel approach for data-driven model reduction of time-dependent parametric partial differential equations. Using a multi-step procedure consisting of proper orthogonal decomposition, dynamic mode decomposition, and manifold interpolation, the proposed approach allows to accurately recover field solutions from a few large-scale simulations. Numerical experiments for the Rayleigh-Bénard cavity problem show the effectiveness of such multi-step procedure in two parametric regimes, i.e., medium and high Grashof number. The latter regime is particularly challenging as it nears the onset of turbulent and chaotic behavior. A major advantage of the proposed method in the context of time-periodic solutions is the ability to recover frequencies that are not present in the sampled data. 

Vertical motion is an important driver of sunlight exposure in aquatic environments, shaping the growth and fate of materials and organisms. We derive a simple model accounting for turbulent depth fluctuations of particles to predict the depth that contributes the most sunlight exposure (effective depth) as well as the single depth that, if measured at one place over time, produces the same total sunlight exposure as a moving particle (functional depth). Field measurements of light and depth in rivers using neutrally buoyant drifters and buoys validate our model. Effective depth varied from 0.1 to 1.5 m below the water surface and was ~ 30% of the overall water depth on average. Functional depth varied from 0.67 to 2.3 m and was ~ 50% of the overall water depth on average. Functional and effective depth are physically based concepts incorporating turbulent motion, spatial variability, and water clarity offering new approaches to characterize light exposure in aquatic environments.

 

We give a systematic self-contained exposition of how to construct geometrically decomposed bases and degrees of freedom in finite element exterior calculus. In particular, we elaborate upon a previously overlooked basis for one of the families of finite element spaces, which is of interest for implementations. Moreover, we give details for the construction of isomorphisms and duality pairings between finite element spaces. These structural results show, for example, how to transfer linear dependencies between canonical spanning sets, or how to derive the degrees of freedom.

 

We address fundamental aspects in the approximation theory of vector-valued finite element methods, using finite element exterior calculus as a unifying framework. We generalize the Clément interpolant and the Scott-Zhang interpolant to finite element differential forms, and we derive a broken Bramble-Hilbert lemma. Our interpolants require only minimal smoothness assumptions and respect partial boundary conditions. This permits us to state local error estimates in terms of the mesh size. Our theoretical results apply to curl-conforming and divergence-conforming finite element methods over simplicial triangulations.