Convergence and Optimality of Adaptive Mixed Methods on Surfaces

In a 1988 article, Dziuk introduced a nodal finite element method for the Laplace-Beltrami equation on 2-surfaces approximated by a piecewise-linear triangulation, initiating a line of research into surface finite element methods (SFEM). Demlow and Dziuk built on the original results, introducing an adaptive method for problems on 2-surfaces, and Demlow later extended the a priori theory to 3-surfaces and higher order elements. In a separate line of research, the Finite Element Exterior Calculus (FEEC) framework has been developed over the last decade by Arnold, Falk and Winther and others as a way to exploit the observation that mixed variational problems can be posed on a Hilbert complex, and Galerkin-type mixed methods can be obtained by solving finite dimensional subproblems. In 2011, Holst and Stern merged these two lines of research by developing a framework for variational crimes in abstract Hilbert complexes, allowing for application of the FEEC framework to problems that violate the subcomplex assumption of Arnold, Falk and Winther. When applied to Euclidean hypersurfaces, this new framework recovers the original a priori results and extends the theory to problems posed on surfaces of arbitrary dimensions. In yet another seemingly distinct line of research, Holst, Mihalik and Szypowski developed a convergence theory for a specific class of adaptive problems in the FEEC framework. Here, we bring these ideas together, showing convergence and optimality of an adaptive finite element method for the mixed formulation of the Hodge Laplacian on hypersurfaces.Comment: 22 pages, no figures. arXiv admin note: substantial text overlap with arXiv:1306.188

The Role of Visual Sensory Performance Outcomes in Concussions: Impact on Concussed Special Operations Forces Combat Soldiers and Possible Implications for the Future of Sports-Related Concussions

The Role of Visual Sensory Performance Outcomes in Concussions: Impact on Concussed Special Operations Forces Combat Soldiers and Possible Implications for the Future of Sports-Related Concussions Clara Soligon, BS, CAT(C) Concordia University, 2022 This thesis details the increased concern towards concussions in athletes and Soldiers as well as the role of visual sensory performance. More studies are showing the consequences, whether short term or long term, of concussions. The symptoms burden and multiple neurocognitive deficits faced by a concussed athlete are getting increasingly recognized by society, as well as healthcare professionals. Studies have shown how concussions can also cause visual sensory performance deficits, even when traditional assessments are normal, and the athletes are cleared to return to play. One big challenge with concussions is the lack of objective measures to diagnose a concussion, as well for medically clearing an athlete or Soldier to return to full activity. Even with the knowledge that visual deficits might be present after a concussion; most traditional assessments do not assess vision due to a lack of unified platform and test availability. Visual sensory performance is important for injury prevention and impact anticipation, as well as assuring peak occupational performance. We assessed US Special Operations Forces combat Soldiers’ visual sensory performance outcomes. Concussions are the most common traumatic injury in the US military since 2000. Visual sensory performance outcome deficits could prevent Soldiers to complete their missions and impact their safety. Finding visual sensory performance outcomes deficits could help prevent an early return to play or return to duty, an increased risk of re-injury as well as help guide rehabilitation in athletes and military alike

The Effects of External Jugular Compression Applied during Head Impact Exposure on Longitudinal Changes in Brain Neuroanatomical and Neurophysiological Biomarkers: A Preliminary Investigation

Objectives: Utilize a prospective in vivo clinical trial to evaluate the potential for mild neck compression applied during head impact exposure to reduce anatomical and physiological biomarkers of brain injury. Methods: This project utilized a prospective randomized controlled trial to evaluate effects of mild jugular vein (neck) compression (collar) relative to controls (no collar) during a competitive hockey season (males; 16.3 ± 1.2 years). The collar was designed to mildly compress the jugular vein bilaterally with the goal to increase intracranial blood volume to reduce risk of brain slosh injury during head impact exposure. Helmet sensors were used to collect daily impact data in excess of 20 g (games and practices) and the primary outcome measures, which included changes in white matter (WM) microstructure, were assessed by diffusion tensor imaging (DTI). Specifically, four DTI measures: fractional anisotropy, mean diffusivity (MD), axial diffusivity, and radial diffusivity (RD) were used in the study. These metrics were analyzed using the tract-based Spatial Statistics (TBSS) approach – a voxel-based analysis. In addition, electroencephalography-derived event-related potentials were used to assess changes in brain network activation (BNA) between study groups. Results: For athletes not wearing the collar, DTI measures corresponding to a disruption of WM microstructure, including MD and RD, increased significantly from pre-season to mid-season (p 0.05). In addition to these anatomical findings, electrophysiological network analysis of the degree of congruence in the network electrophysiological activation pattern demonstrated concomitant changes in brain network dynamics in the non-collar group only (p < 0.05). Similar to the DTI findings, the increased change in BNA score in the non-collar relative to the collar group was statistically significant (p < 0.01). Changes in DTI outcomes were also directly correlated with altered brain network dynamics (r = 0.76; p < 0.05) as measured by BNA. Conclusion: Group differences in the longitudinal changes in both neuroanatomical and electrophysiological measures, as well as the correlation between the measures, provide initial evidence indicating that mild jugular vein compression may have reduced alterations in the WM response to head impacts during a competitive hockey season. The data indicate sport-related alterations in WM microstructure were ameliorated by application of jugular compression during head impact exposure. These results may lead to a novel line of research inquiry to evaluate the effects of protecting the brain from sports-related head impacts via optimized intracranial fluid dynamics

Adaptive Methods in the Finite Element Exterior Calculus Framework /

In this thesis we explore convergence theory for adaptive mixed finite element methods. In particular, we introduce an a posteriori error-indicator, and prove convergence and optimality results for the mixed formulation of the Hodge Laplacian posed on domains of arbitrary dimensionality and topology in R/n. After developing this framework, we introduce a new algorithm and extend our theory and results to problems posed on Euclidean hypersurfaces. We begin by introducing the finite element exterior calculus framework, which is the key tool allowing us to address the convergence proofs in such generality. This introduction focuses on the fundamentals of the well- developed a priori theory and the results needed to extend the core of this theory to problems posed on surfaces. A basic set of results needed to develop adaptivity in this framework is also established. We then introduce an adaptive algorithm, and show convergence using this infrastructure as a tool to generalize existing finite element theory. The algorithm is then shown to be computationally optimal through a series of complexity analysis arguments. Finally, a second algorithm is introduced for problems posed on surfaces, and our original convergence and optimality results are extended using properties of specific geometric maps between surface

CONVERGENCE AND OPTIMALITY OF ADAPTIVE METHODS IN THE FINITE ELEMENT EXTERIOR CALCULUS FRAMEWORK

ABSTRACT. Finite Element Exterior Calculus (FEEC) was developed by Arnold, Falk, Winther and others over the last decade to exploit the observation that mixed variational problems can be posed on a Hilbert Complex, and Galerkin-type mixed methods can then be obtained by solving finite-dimensional subcomplex problems. Stability and consistency of the resulting methods then follow directly from the framework by establishing the existence of operators connecting the Hilbert complex with its subcomplex, giving a essentially a “recipe ” for well-behaved methods. In 2012, Demlow and Hirani developed a posteriori error indicators for driving adaptive methods in the FEEC framework. While adaptive techniques have been used successfully with mixed methods for years, convergence theory for such techniques has not been fully developed. The main difficulty is lack of error orthogonality. In 2009, Chen, Holst, and Xu established convergence and optimality of an adaptive mixed finite element method for the Poisson equation (the Hodge-Laplace problem fork = n = 2) on simply connected polygonal domains in two dimensions. Their argument used a type of quasi-orthogonality result, exploiting the fact that the error was orthogonal to the divergence free subspace, while the part of the erro