On Basis Constructions in Finite Element Exterior Calculus

We give a systematic and self-contained account of the construction of 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 give a new derivation of the degrees of freedom

Symmetry and Invariant Bases in Finite Element Exterior Calculus

We study symmetries of bases and spanning sets in finite element exterior calculus using representation theory. The group of affine symmetries of a simplex is isomorphic to a permutation group and represented on simplicial finite element spaces by the pullback action. We want to know which vector-valued finite element spaces have bases that are invariant under permutation of vertex indices. We determine a natural notion of invariance and sufficient conditions on the dimension and polynomial degree for the existence of invariant bases. We conjecture that these conditions are necessary too. We utilize Djokovic and Malzan's classification of monomial irreducible representations of the symmetric group and use symmetries of the geometric decomposition and canonical isomorphisms of the finite element spaces. Invariant bases are constructed in dimensions two and three for different spaces of finite element differential forms.Comment: 27 pages. Submitte

Higher-order chain rules for tensor fields, generalized Bell polynomials, and estimates in Orlicz-Sobolev-Slobodeckij and bounded variation spaces

We describe higher-order chain rules for multivariate functions and tensor fields. We estimate Sobolev-Slobodeckij norms, Musielak-Orlicz norms, and the total variation seminorms of the higher derivatives of tensor fields after a change of variables and determine sufficient regularity conditions for the coordinate change. We also introduce a novel higher-order chain rule for composition chains of multivariate functions that is described via nested set partitions and generalized Bell polynomials; it is a natural extension of the Fa\`a di Bruno formula. Our discussion uses the coordinate-free language of tensor calculus and includes Fr\'echet-differentiable mappings between Banach spaces.Comment: Submitte

Smoothed projections over manifolds in finite element exterior calculus

We develop commuting finite element projections over smooth Riemannian manifolds. This extension of finite element exterior calculus establishes the stability and convergence of finite element methods for the Hodge-Laplace equation on manifolds. The commuting projections use localized mollification operators, building upon a classical construction by de Rham. These projections are uniformly bounded on Lebesgue spaces of differential forms and map onto intrinsic finite element spaces defined with respect to an intrinsic smooth triangulation of the manifold. We analyze the Galerkin approximation error. Since practical computations use extrinsic finite element methods over approximate computational manifolds, we also analyze the geometric error incurred.Comment: Submitted. 31 page

Corn (Zea mays L.) seeding rate optimization in Iowa, USA

Collecting soil, topography, and yield information has become more feasible and reliable with advancements in precision technologies. Combined with the accessibility of precision technologies and services to farmers, there has been increased interest and ability to make site-specific crop management decisions. The objective of this research was to develop procedures to optimize corn seeding rates and maximize yield using soil and topographic parameters. Experimental treatments included five seeding rates (61 750; 74 100; 86 450; 98 800; and 111 150 seeds ha−1) in a randomized complete block design in three central Iowa fields from 2012 to 2014 (nine site-years). Soil samples were analyzed for available phosphorus (Olsen method), exchangeable potassium (ammonium-acetate method), pH, soil organic matter (SOM), cation exchange capacity (CEC), and texture. Topographic data (in-field elevation, slope, aspect, and curvature) were determined from publically available light detection and ranging data. In four site-years, no interaction occurred between seeding rate and the descriptive variables. Three of the site-years resulted in a negative linear seeding rate response which made it impossible to determine an optimum seeding rate above the lowest seeding rate treatment. The seeding rate optimization process in five site-years resulted in seeding rate by variable interactions; four site-years had a single seeding rate by variable interaction (pH, in-field elevation, or curvature) and one site-year had three seeding rate by variable interactions (pH, CEC, and SOM). Meaningful seeding rate optimizations occurred in only three of nine site-years. There was not a consistent descriptive variable interaction with seeding rate as a result of weather variability

Corn (\u3ci\u3eZea mays L.\u3c/i\u3e) seeding rate optimization in Iowa, USA

Collecting soil, topography, and yield information has become more feasible and reliable with advancements in precision technologies. Combined with the accessibility of precision technologies and services to farmers, there has been increased interest and ability to make site-specific crop management decisions. The objective of this research was to develop procedures to optimize corn seeding rates and maximize yield using soil and topographic parameters. Experimental treatments included five seeding rates (61 750; 74 100; 86 450; 98 800; and 111 150 seeds ha-1) in a randomized complete block design in three central Iowa fields from 2012 to 2014 (nine site-years). Soil samples were analyzed for available phosphorus (Olsen method), exchangeable potassium (ammonium-acetate method), pH, soil organic matter (SOM), cation exchange capacity (CEC), and texture. Topographic data (in-field elevation, slope, aspect, and curvature) were determined from publically available light detection and ranging data. In four site-years, no interaction occurred between seeding rate and the descriptive variables. Three of the site-years resulted in a negative linear seeding rate response which made it impossible to determine an optimum seeding rate above the lowest seeding rate treatment. The seeding rate optimization process in five site-years resulted in seeding rate by variable interactions; four site-years had a single seeding rate by variable interaction (pH, in-field elevation, or curvature) and one site-year had three seeding rate by variable interactions (pH, CEC, and SOM). Meaningful seeding rate optimizations occurred in only three of nine site-years. There was not a consistent descriptive variable interaction with seeding rate as a result of weather variability. Referenc

Corn (\u3ci\u3eZea mays L.\u3c/i\u3e) seeding rate optimization in Iowa, USA

Collecting soil, topography, and yield information has become more feasible and reliable with advancements in precision technologies. Combined with the accessibility of precision technologies and services to farmers, there has been increased interest and ability to make site-specific crop management decisions. The objective of this research was to develop procedures to optimize corn seeding rates and maximize yield using soil and topographic parameters. Experimental treatments included five seeding rates (61 750; 74 100; 86 450; 98 800; and 111 150 seeds ha-1) in a randomized complete block design in three central Iowa fields from 2012 to 2014 (nine site-years). Soil samples were analyzed for available phosphorus (Olsen method), exchangeable potassium (ammonium-acetate method), pH, soil organic matter (SOM), cation exchange capacity (CEC), and texture. Topographic data (in-field elevation, slope, aspect, and curvature) were determined from publically available light detection and ranging data. In four site-years, no interaction occurred between seeding rate and the descriptive variables. Three of the site-years resulted in a negative linear seeding rate response which made it impossible to determine an optimum seeding rate above the lowest seeding rate treatment. The seeding rate optimization process in five site-years resulted in seeding rate by variable interactions; four site-years had a single seeding rate by variable interaction (pH, in-field elevation, or curvature) and one site-year had three seeding rate by variable interactions (pH, CEC, and SOM). Meaningful seeding rate optimizations occurred in only three of nine site-years. There was not a consistent descriptive variable interaction with seeding rate as a result of weather variability. Referenc

Local Finite Element Approximation of Sobolev Differential Forms

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 Cl\'ement 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.Comment: 22 pages. Comments welcom