Andrew Gillette

Alexander Rand, Andrew Gillette, Chandrajit Bajaj

We introduce a finite element construction for use on the class of convex, planar polygons and show it obtains a quadratic error convergence estimate. On a convex n-gon satisfying simple geometric criteria, our construction produces 2n basis functions, associated in a Lagrange-like fashion to each vertex and each edge midpoint, by transforming and combining a set of n(n+1)/2 basis functions known to obtain quadratic convergence. The technique broadens the scope of the so-called `serendipity' elements, previously studied only for quadrilateral and regular hexahedral meshes, by employing the theory of generalized barycentric coordinates. Uniform `a priori' error estimates are established over the class of convex quadrilaterals with bounded aspect ratio as well as over the class of generic convex planar polygons satisfying additional shape regularity conditions to exclude large interior angles and short edges. Numerical evidence is provided on a trapezoidal quadrilateral mesh, previously not amenable to serendipity constructions, and applications to adaptive meshing are discussed.

Andrew Gillette, Alexander Rand, Chandrajit Bajaj

We prove the optimal convergence estimate for first order interpolants used in finite element methods based on three major approaches for generalizing barycentric interpolation functions to convex planar polygonal domains. The Wachspress approach explicitly constructs rational functions, the Sibson approach uses Voronoi diagrams on the vertices of the polygon to define the functions, and the Harmonic approach defines the functions as the solution of a PDE. We show that given certain conditions on the geometry of the polygon, each of these constructions can obtain the optimal convergence estimate. In particular, we show that the well-known maximum interior angle condition required for interpolants over triangles is still required for Wachspress functions but not for Sibson functions.

Andrew Gillette, Alexander Rand, Chandrajit Bajaj

We combine theoretical results from polytope domain meshing, generalized barycentric coordinates, and finite element exterior calculus to construct scalar- and vector-valued basis functions for conforming finite element methods on generic convex polytope meshes in dimensions 2 and 3. Our construction recovers well-known bases for the lowest order Nédélec, Raviart-Thomas, and Brezzi-Douglas-Marini elements on simplicial meshes and generalizes the notion of Whitney forms to non-simplicial convex polygons and polyhedra. We show that our basis functions lie in the correct function space with regards to global continuity and that they reproduce the requisite polynomial differential forms described by finite element exterior calculus. We present a method to count the number of basis functions required to ensure these two key properties.

Andrew Gillette, Brendan Keith, Socratis Petrides

In this work, we revisit the marking decisions made in the standard adaptive finite element method (AFEM). Experience shows that a naïve marking policy leads to inefficient use of computational resources for adaptive mesh refinement (AMR). Consequently, using AFEM in practice often involves ad-hoc or time-consuming offline parameter tuning to set appropriate parameters for the marking subroutine. To address these practical concerns, we recast AMR as a Markov decision process in which refinement parameters can be selected on-the-fly at run time, without the need for pre-tuning by expert users. In this new paradigm, the refinement parameters are also chosen adaptively via a marking policy that can be optimized using methods from reinforcement learning. We use the Poisson equation to demonstrate our techniques on $h$- and $hp$-refinement benchmark problems, and our experiments suggest that superior marking policies remain undiscovered for many classical AFEM applications. Furthermore, an unexpected observation from this work is that marking policies trained on one family of PDEs are sometimes robust enough to perform well on problems far outside the training family. For illustration, we show that a simple $hp$-refinement policy trained on 2D domains with only a single re-entrant corner can be deployed on far more complicated 2D domains, and even 3D domains, without significant performance loss. For reproduction and broader adoption, we accompany this work with an open-source implementation of our methods.

Alexander Rand, Andrew Gillette, Chandrajit Bajaj

In a similar fashion to estimates shown for Harmonic, Wachspress, and Sibson coordinates in [Gillette et al., AiCM, doi:10.1007/s10444-011-9218-z], we prove interpolation error estimates for the mean value coordinates on convex polygons suitable for standard finite element analysis. Our analysis is based on providing a uniform bound on the gradient of the mean value functions for all convex polygons of diameter one satisfying certain simple geometric restrictions. This work makes rigorous an observed practical advantage of the mean value coordinates: unlike Wachspress coordinates, the gradients of the mean value coordinates do not become large as interior angles of the polygon approach pi.

Michael S. Floater, Andrew Gillette

Using the notion of multivariate lower set interpolation, we construct nodal basis functions for the serendipity family of finite elements, of any order and any dimension. For the purpose of computation, we also show how to express these functions as linear combinations of tensor-product polynomials.

Andrew Gillette, Michael Holst, Yunrong Zhu

Arnold, Falk, and Winther [Bull. Amer. Math. Soc. 47 (2010), 281--354] showed that mixed variational problems, and their numerical approximation by mixed methods, could be most completely understood using the ideas and tools of Hilbert complexes. This led to the development of the Finite Element Exterior Calculus (FEEC) for a large class of linear elliptic problems. More recently, Holst and Stern [Found. Comp. Math. 12:3 (2012), 263--293 and 363--387] extended the FEEC framework to semi-linear problems, and to problems containing variational crimes, allowing for the analysis and numerical approximation of linear and nonlinear geometric elliptic partial differential equations on Riemannian manifolds of arbitrary spatial dimension, generalizing surface finite element approximation theory. In this article, we develop another distinct extension to the FEEC, namely to parabolic and hyperbolic evolution systems, allowing for the treatment of geometric and other evolution problems. Our approach is to combine the recent work on the FEEC for elliptic problems with a classical approach to solving evolution problems via semi-discrete finite element methods, by viewing solutions to the evolution problem as lying in time-parameterized Hilbert spaces (or Bochner spaces). Building on classical approaches by Thomee for parabolic problems and Geveci for hyperbolic problems, we establish a priori error estimates for Galerkin FEM approximation in the natural parametrized Hilbert space norms. In particular, we recover the results of Thomee and Geveci for two-dimensional domains and lowest-order mixed methods as special cases, effectively extending their results to arbitrary spatial dimension and to an entire family of mixed methods. We also show how the Holst and Stern framework allows for extensions of these results to certain semi-linear evolution problems.

Andrew Gillette, Tyler Kloefkorn

We introduce the family of trimmed serendipity finite element differential form spaces, defined on cubical meshes in any number of dimensions, for any polynomial degree, and for any form order. The relation between the trimmed serendipity family and the (non-trimmed) serendipity family developed by Arnold and Awanou [Math. Comp. 83(288) 2014] is analogous to the relation between the trimmed and (non-trimmed) polynomial finite element differential form families on simplicial meshes from finite element exterior calculus. We provide degrees of freedom in the general setting and prove that they are unisolvent for the trimmed serendipity spaces. The sequence of trimmed serendipity spaces with a fixed polynomial order r provides an explicit example of a system described by Christiansen and Gillette [ESAIM:M2AN 50(3) 2016], namely, a minimal compatible finite element system on squares or cubes containing order r-1 polynomial differential forms.

Andrew Gillette, Kaibo Hu, Shuo Zhang

We propose two general operations on finite element differential complexes on cubical meshes that can be used to construct and analyze sequences of "nonstandard" convergent methods. The first operation, called DoF-transfer, moves edge degrees of freedom to vertices in a way that reduces global degrees of freedom while increasing continuity order at vertices. The second operation, called serendipity, eliminates interior bubble functions and degrees of freedom locally on each element without affecting edge degrees of freedom. These operations can be used independently or in tandem to create nonstandard complexes that incorporate Hermite, Adini and "trimmed-Adini" elements. We show that the resulting elements provide convergent, non-conforming methods for problems requiring stronger regularity and satisfy a discrete Korn inequality. We discuss potential benefits of applying these elements to Stokes, biharmonic and elasticity problems.

Andrew Gillette, Tyler Kloefkorn, Victoria Sanders

Many conforming finite elements on squares and cubes are elegantly classified into families by the language of finite element exterior calculus and presented in the Periodic Table of the Finite Elements. Use of these elements varies, based principally on the ease or difficulty in finding a "computational basis" of shape functions for element families. The tensor product family, $Q^-_rΛ^k$, is most commonly used because computational basis functions are easy to state and implement. The trimmed and non-trimmed serendipity families, $S^-_rΛ^k$ and $S_rΛ^k$ respectively, are used less frequently because they are newer to the community and, until now, lacked a straightforward technique for computational basis construction. This represents a missed opportunity for computational efficiency as the serendipity elements in general have fewer degrees of freedom than elements of equivalent accuracy from the tensor product family. Accordingly, in pursuit of easy adoption of the serendipity families, we present complete lists of computational bases for both serendipity families, for any order $r\geq 1$ and for any differential form order $0\leq k\leq n$, for problems in dimension $n=2$ or $3$. The bases are defined via shared subspace structures, allowing easy comparison of elements across families. We use and include code in SageMath to find, list, and verify these computational basis functions.

Andrew Gillette

We introduce new Hermite-style and Bernstein-style geometric decompositions of the cubic order serendipity finite element spaces $S_3(I^2)$ and $S_3(I^3)$, as defined in the recent work of Arnold and Awanou [Found. Comput. Math. 11 (2011), 337--344]. The serendipity spaces are substantially smaller in dimension than the more commonly used bicubic and tricubic Hermite tensor product spaces - 12 instead of 16 for the square and 32 instead of 64 for the cube - yet are still guaranteed to obtain cubic order \textit{a priori} error estimates in $H^1$ norm when used in finite element methods. The basis functions we define have a canonical relationship both to the finite element degrees of freedom as well as to the geometry of their graphs; this means the bases may be suitable for applications employing isogeometric analysis where domain geometry and functions supported on the domain are described by the same basis functions. Moreover, the basis functions are linear combinations of the commonly used bicubic and tricubic polynomial Bernstein or Hermite basis functions, allowing their rapid incorporation into existing finite element codes.

Andrew Gillette, Craig Gross, Ken Plackowski

While the use of finite element methods for the numerical approximation of eigenvalues is a well-studied problem, the use of serendipity elements for this purpose has received little attention in the literature. We show by numerical experiments that serendipity elements, which are defined on a square reference geometry, can attain the same order of accuracy as their tensor product counterparts while using dramatically fewer degrees of freedom. In some cases, the serendipity method uses only half as many basis functions as the tensor product method while still producing the same numerical approximation of an eigenvalue. To encourage the further use and study of serendipity elements, we provide a table of serendipity basis functions for low order cases and a Mathematica file that can be used to generate the basis functions for higher order cases.

Jennifer Zvonek, Andrew Gillette

An implicit neural representation (INR) is a neural network that approximates a spatiotemporal function. Many memory-intensive visualization tasks, including modern 4D CT scanning methods, represent data natively as INRs. While INRs are prized for being more memory-efficient than traditional data stored on a lattice, many visualization tasks still require discretization to a regular grid. We present PruningAMR, an algorithm that builds a mesh with resolution adapted to geometric features encoded by the INR. To identify these geometric features, we use an interpolative decomposition pruning method on the weight matrices of the INR. The resulting pruned network is used to guide adaptive mesh refinement, enabling automatic mesh generation tailored to the underlying resolution of the function. Starting from a pre-trained INR--without access to its training data--we produce a variable resolution visualization with substantial memory savings.

Tyler Chang, Andrew Gillette, Romit Maulik

Effective verification and validation techniques for modern scientific machine learning workflows are challenging to devise. Statistical methods are abundant and easily deployed, but often rely on speculative assumptions about the data and methods involved. Error bounds for classical interpolation techniques can provide mathematically rigorous estimates of accuracy, but often are difficult or impractical to determine computationally. In this work, we present a best-of-both-worlds approach to verifiable scientific machine learning by demonstrating that (1) multiple standard interpolation techniques have informative error bounds that can be computed or estimated efficiently; (2) comparative performance among distinct interpolants can aid in validation goals; (3) deploying interpolation methods on latent spaces generated by deep learning techniques enables some interpretability for black-box models. We present a detailed case study of our approach for predicting lift-drag ratios from airfoil images. Code developed for this work is available in a public Github repository.

Andrew Gillette

Using the language of finite element exterior calculus, we define two families of $H^1$-conforming finite element spaces over pyramids with a parallelogram base. The first family has matching polynomial traces with tensor product elements on the base while the second has matching polynomial traces with serendipity elements on the base. The second family is new to the literature and provides a robust approach for linking between Lagrange elements on tetrahedra and serendipity elements on affinely-mapped cubes while preserving continuity and approximation properties. We define shape functions and degrees of freedom for each family and prove unisolvence and polynomial reproduction results.

Chandrajit Bajaj, Andrew Gillette, Qin Zhang

Current mesh reduction techniques, while numerous, all primarily reduce mesh size by successive element deletion (e.g. edge collapses) with the goal of geometric and topological feature preservation. The choice of geometric error used to guide the reduction process is chosen independent of the function the end user aims to calculate, analyze, or adaptively refine. In this paper, we argue that such a decoupling of structure from function modeling is often unwise as small changes in geometry may cause large changes in the associated function. A stable approach to mesh decimation, therefore, ought to be guided primarily by an analysis of functional sensitivity, a property dependent on both the particular application and the equations used for computation (e.g. integrals, derivatives, or integral/partial differential equations). We present a methodology to elucidate the geometric sensitivity of functionals via two major functional discretization techniques: Galerkin finite element and discrete exterior calculus. A number of examples are given to illustrate the methodology and provide numerical examples to further substantiate our choices.