Numerical Experiments for Darcy Flow on a Surface Using Mixed Exterior Calculus Methods

There are very few results on mixed finite element methods on surfaces. A theory for the study of such methods was given recently by Holst and Stern, using a variational crimes framework in the context of finite element exterior calculus. However, we are not aware of any numerical experiments where mixed finite elements derived from discretizations of exterior calculus are used for a surface domain. This short note shows results of our preliminary experiments using mixed methods for Darcy flow (hence scalar Poisson's equation in mixed form) on surfaces. We demonstrate two numerical methods. One is derived from the primal-dual Discrete Exterior Calculus and the other from lowest order finite element exterior calculus. The programming was done in the language Python, using the PyDEC package which makes the code very short and easy to read. The qualitative convergence studies seem to be promising.Comment: 14 pages, 11 figure

Least Squares Ranking on Graphs

Given a set of alternatives to be ranked, and some pairwise comparison data, ranking is a least squares computation on a graph. The vertices are the alternatives, and the edge values comprise the comparison data. The basic idea is very simple and old: come up with values on vertices such that their differences match the given edge data. Since an exact match will usually be impossible, one settles for matching in a least squares sense. This formulation was first described by Leake in 1976 for rankingfootball teams and appears as an example in Professor Gilbert Strang's classic linear algebra textbook. If one is willing to look into the residual a little further, then the problem really comes alive, as shown effectively by the remarkable recent paper of Jiang et al. With or without this twist, the humble least squares problem on graphs has far-reaching connections with many current areas ofresearch. These connections are to theoretical computer science (spectral graph theory, and multilevel methods for graph Laplacian systems); numerical analysis (algebraic multigrid, and finite element exterior calculus); other mathematics (Hodge decomposition, and random clique complexes); and applications (arbitrage, and ranking of sports teams). Not all of these connections are explored in this paper, but many are. The underlying ideas are easy to explain, requiring only the four fundamental subspaces from elementary linear algebra. One of our aims is to explain these basic ideas and connections, to get researchers in many fields interested in this topic. Another aim is to use our numerical experiments for guidance on selecting methods and exposing the need for further development.Comment: Added missing references, comparison of linear solvers overhauled, conclusion section added, some new figures adde

Energy Conserving Higher Order Mixed Finite Element Discretizations of Maxwell's Equations

We study a system of Maxwell's equations that describes the time evolution of electromagnetic fields with an additional electric scalar variable to make the system amenable to a mixed finite element spatial discretization. We demonstrate stability and energy conservation for the variational formulation of this Maxwell's system. We then discuss two implicit, energy conserving schemes for its temporal discretization: the classical Crank-Nicholson scheme and an implicit leapfrog scheme. We next show discrete stability and discrete energy conservation for the semi-discretization using these two time integration methods. We complete our discussion by showing that the error for the full discretization of the Maxwell's system with each of the two implicit time discretization schemes and with spatial discretization through a conforming sequence of de Rham finite element spaces converges quadratically in the step size of the time discretization and as an appropriate polynomial power of the mesh parameter in accordance with the choice of approximating polynomial spaces. Our results for the Crank-Nicholson method are generally well known but have not been demonstrated for this Maxwell's system. Our implicit leapfrog scheme is a new method to the best of our knowledge and we provide a complete error analysis for it. Finally, we show computational results to validate our theoretical claims using linear and quadratic Whitney forms for the finite element discretization for some model problems in two and three spatial dimensions

Hodge Laplacians on simplicial meshes and graphs

We present in this dissertation some developments in the discretizations of exterior calculus for problems posed on simplicial discretization (meshes) of geometric manifolds and analogous problems on abstract simplicial complexes. We are primarily interested in discretizations of elliptic type partial differential equations, and our model problem is the Hodge Laplacian Poisson problem on differential k-forms on n dimensional manifolds. One of our major contributions in this work is the computational quantification of the solution using the weak mixed formulation of this problem on simplicial meshes using discrete exterior calculus (DEC), and its comparisons with the solution due to a different discretization framework, namely, finite element exterior calculus (FEEC). Consequently, our important computational result is that the solution of the Poisson problem on different manifolds in two- and three-dimensions due to DEC recovers convergence properties on many sequences of refined meshes similar to that of FEEC. We also discuss some potential attempts for showing this convergence theoretically. In particular, we demonstrate that a certain formulation of a variational crimes approach that can be used for showing convergence for a generalized FEEC may not be directly applicable to DEC convergence in its current formulation. In order to perform computations using DEC, a key development that we present is exhibiting sign rules that allow for the computation of the discrete Hodge star operators in DEC on Delaunay meshes in a piecewise manner. Another aspect of computationally solving the Poisson problem using the mixed formulation with either DEC or FEEC requires knowing the solution to the corresponding Laplace's problem, namely, the harmonics. We present a least squares method for computing a basis for the space of such discrete harmonics via their isomorphism to cohomology. We also provide some numerics to quantify the efficiency of this solution in comparison with previously known methods. Finally, we demonstrate an application to obtain the ranking of pairwise comparison data. We model this data as edge weights on graphs with 3-cliques included and perform its Hodge decomposition by solving two least squares problems. An outcome of this exploration is also providing some computational evidence that algebraic multigrid linear solvers for the resulting linear systems on Erdős-Rényi random graphs and on Barabási-Albert graphs do not perform very well in comparison with iterative Krylov solvers

Quantifying macrostructures in viscoelastic sub-diffusive flows

We present a theory to quantify the formation of spatiotemporal macrostructures (or the non-homogeneous regions of high viscosity at moderate to high fluid inertia) for viscoelastic sub-diffusive flows, by introducing a mathematically consistent decomposition of the polymer conformation tensor, into the so-called structure tensor. Our approach bypasses an inherent problem in the standard arithmetic decomposition, namely, the fluctuating conformation tensor fields may not be positive definite and hence, do not retain their physical meaning. Using well-established results in matrix analysis, the space of positive definite matrices is transformed into a Riemannian manifold by defining and constructing a geodesic via the inner product on its tangent space. This geodesic is utilized to define three scalar invariants of the structure tensor, which do not suffer from the caveats of the regular invariants (such as trace and determinant) of the polymer conformation tensor. First, we consider the problem of formulating perturbative expansions of the structure tensor using the geodesic, which is consistent with the Riemannian manifold geometry. A constraint on the maximum time, during which the evolution of the perturbative solution can be well approximated by linear theory along the Euclidean manifold, is found. Finally, direct numerical simulations of the viscoelastic sub-diffusive channel flows (where the stress-constitutive law is obtained via coarse-graining the polymer relaxation spectrum at finer scale, Chauhan et. al., Phys. Fluids, DOI: 10.1063/5.0174598 (2023)), underscore the advantage of using these invariants in effectively quantifying the macrostructures.Comment: 19 pages, 2 figure

Delaunay Hodge Star

We define signed dual volumes at all dimensions for circumcentric dual meshes. We show that for pairwise Delaunay triangulations with mild boundary assumptions these signed dual volumes are positive. This allows the use of such Delaunay meshes for Discrete Exterior Calculus (DEC) because the discrete Hodge star operator can now be correctly defined for such meshes. This operator is crucial for DEC and is a diagonal matrix with the ratio of primal and dual volumes along the diagonal. A correct definition requires that all entries be positive. DEC is a framework for numerically solving differential equations on meshes and for geometry processing tasks and has had considerable impact in computer graphics and scientific computing. Our result allows the use of DEC with a much larger class of meshes than was previously considered possible.Comment: Corrected error in Figure 1 (columns 3 and 4) and Figure 6 and a formula error in Section 2. All mathematical statements (theorems and lemmas) are unchanged. The previous arXiv version v3 (minus the Appendix) appeared in the journal Computer-Aided Desig

Using the MitoB method to assess levels of reactive oxygen species in ecological studies of oxidative stress

In recent years evolutionary ecologists have become increasingly interested in the effects of reactive oxygen species (ROS) on the life-histories of animals. ROS levels have mostly been inferred indirectly due to the limitations of estimating ROS from in vitro methods. However, measuring ROS (hydrogen peroxide, H2O2) content in vivo is now possible using the MitoB probe. Here, we extend and refine the MitoB method to make it suitable for ecological studies of oxidative stress using the brown trout Salmo trutta as model. The MitoB method allows an evaluation of H2O2 levels in living organisms over a timescale from hours to days. The method is flexible with regard to the duration of exposure and initial concentration of the MitoB probe, and there is no transfer of the MitoB probe between fish. H2O2 levels were consistent across subsamples of the same liver but differed between muscle subsamples and between tissues of the same animal. The MitoB method provides a convenient method for measuring ROS levels in living animals over a significant period of time. Given its wide range of possible applications, it opens the opportunity to study the role of ROS in mediating life history trade-offs in ecological settings

Introducing a distributed unstructured mesh into gyrokinetic particle-in-cell code, XGC

XGC has shown good scalability for large leadership supercomputers. The current production version uses a copy of the entire unstructured finite element mesh on every MPI rank. Although an obvious scalability issue if the mesh sizes are to be dramatically increased, the current approach is also not optimal with respect to data locality of particles and mesh information. To address these issues we have initiated the development of a distributed mesh PIC method. This approach directly addresses the base scalability issue with respect to mesh size and, through the use of a mesh entity centric view of the particle mesh relationship, provides opportunities to address data locality needs of many core and GPU supported heterogeneous systems. The parallel mesh PIC capabilities are being built on the Parallel Unstructured Mesh Infrastructure (PUMI) [1]. The presentation will first overview the form of mesh distribution used and indicate the structures and functions used to support the mesh, the particles and their interaction. Attention will then focus on the node-level optimizations being carried out to ensure performant operation of all PIC operations on the distributed mesh

First-principles investigation of the atomic and electronic structure and magnetic moments in gold nanoclusters

We determine the structure, energetics, and emerging magnetic propertiesof Aun clusters using first-principles plane-wave density functional theory. We compare and contrast the findings obtained with and without spin-orbit interaction in the gold clusters of sizes down to 0.5-1 nm. The shapes are chosen to be representative of spherical gold nanoparticles: (a) Au<SUB>13</SUB> and Au<SUB>12</SUB>, with cuboctahedral, icosahedral, and decahedral structures, and (b) Au<SUB>25</SUB> and Au<SUB>24</SUB>, with truncated octahedral structures. We find that the trends in the binding energies are unaltered with the choice of the exchange correlation (LDA or GGA) or the inclusion of relativistic effects. The cuboctahedral Au<SUB>13</SUB> is found to have the lowest binding energy compared with others at a given level of theory. Within the scalar relativistic description, these gold clusters exhibit a wide variety of magnetic moments: the stability and magnetic properties can be readily understood in terms of degeneracies of the HOMO and LUMO levels. Further, there is evidence of Jahn-Teller activity in the cases of cuboctahedral Au<SUB>12</SUB> and Au<SUB>13</SUB> that leads to structural distortion, inducing magnetism in the 12-atom cluster. Analysis of electronic states with projection on atomic orbitals for the scalar relativistic case shows that the magnetism in these gold clusters has an sp rather than the otherwise believed s character. By employing a fully relativistic description with inclusion of spin-orbit interaction and noncollinear magnetization, the magnetic moment in the icosahedral-shaped clusters is found to reduce substantially and that in the 12- and 13-atom clusters with a cuboctahedral structure becomes vanishingly small. The loss of magnetism in Au<SUB>12</SUB> and Au<SUB>13</SUB> appears to originate from the splitting of degenerate HOMO states in these clusters and an overlap of the d and sp states, whereas the magnetic moment of around 1 μ<SUB>B</SUB> in Au<SUB>25</SUB> is mainly caused by the s states of the central atom