Numerical Aspects of the Coefficient Computation for LMMs

The numerical solution of Boundary Value Problems usually requires the use of an adaptive mesh selection strategy. For this reason, when a Linear Multistep Method is considered, a dynamic computation of its coefficients is necessary. This leads to solve linear systems which can be expressed in different forms, depending on the polynomial basis used to impose the order conditions. In this paper, we compare the accuracy of the numerically computed coefficients for three different formulations. For all the considered cases Vandermonde systems on general abscissae are involved and they are always solved by the Bj \u308rk-Pereyra algorithm. An adaptation of the forward error analysis given in [8, 9] is proposed whose significance is confirmed by the numerical results

Cubature rules based on bivariate spline quasi-interpolation for weakly singular integrals

In this paper we present a new class of cubature rules with the aim of accurately integrating weakly singular double integrals. In particular we focus on those integrals coming from the discretization of Boundary Integral Equations for 3D Laplace boundary value problems, using a collocation method within the Isogeometric Analysis paradigm. In such setting the regular part of the integrand can be defined as the product of a tensor product B-spline and a general function. The rules are derived by using first the spline quasi-interpolation approach to approximate such function and then the extension of a well known algorithm for spline product to the bivariate setting. In this way efficiency is ensured, since the locality of any spline quasi-interpolation scheme is combined with the capability of an ad--hoc treatment of the B-spline factor. The numerical integration is performed on the whole support of the B-spline factor by exploiting inter-element continuity of the integrand

IgA-BEM for 3D Helmholtz problems using conforming and non-conforming multi-patch discretizations and B-spline tailored numerical integration

An Isogeometric Boundary Element Method (IgA-BEM) is considered for the numerical solution of Helmholtz problems on 3D bounded or unbounded domains, admitting a smooth multi-patch representation of their finite boundary surface. The discretization spaces are formed by C0 inter-patch continuous functional spaces whose restriction to a patch simplifies to the span of tensor product B-splines composed with the given patch NURBS parameterization. Both conforming and non-conforming spaces are allowed, so that local refinement is possible at the patch level. For regular and singular integration, the proposed model utilizes a numerical procedure defined on the support of each trial B-spline function, which makes possible a function-by-function implementation of the matrix assembly phase. Spline quasi-interpolation is the common ingredient of all the considered quadrature rules; in the singular case it is combined with a B-spline recursion over the spline degree and with a singularity extraction technique, extended to the multi-patch setting for the first time. A threshold selection strategy is proposed to automatically distinguish between nearly singular and regular integrals. The non-conforming C0 joints between spline spaces on different patches are implemented as linear constraints based on knot removal conditions, and do not require a hierarchical master-slave relation between neighbouring patches. Numerical examples on relevant benchmarks show that the expected convergence orders are achieved with uniform discretization and a small number of uniformly spaced quadrature nodes

Time-dependent recovery of brain hypometabolism in neuro-COVID-19 patients

Purpose We evaluated brain metabolic dysfunctions and associations with neurological and biological parameters in acute, subacute and chronic COVID-19 phases to provide deeper insights into the pathophysiology of the disease.Methods Twenty-six patients with neurological symptoms (neuro-COVID-19) and [F-18]FDG-PET were included. Seven patients were acute (< 1 month (m) after onset), 12 subacute (4 >= 1-m, 4 >= 2-m and 4 >= 3-m) and 7 with neuro-post-COVID-19 (3 >= 5-m and 4 >= 7-9-m). One patient was evaluated longitudinally (acute and 5-m). Brain hypo- and hypermetabolism were analysed at single-subject and group levels. Correlations between severity/extent of brain hypo- and hypermetabolism and biological (oxygen saturation and C-reactive protein) and clinical variables (global cognition and Body Mass Index) were assessed.Results The "fronto-insular cortex" emerged as the hypometabolic hallmark of neuro-COVID-19. Acute patients showed the most severe hypometabolism affecting several cortical regions. Three-m and 5-m patients showed a progressive reduction of hypometabolism, with limited frontal clusters. After 7-9 months, no brain hypometabolism was detected. The patient evaluated longitudinally showed a diffuse brain hypometabolism in the acute phase, almost recovered after 5 months. Brain hypometabolism correlated with cognitive dysfunction, low blood saturation and high inflammatory status. Hypermetabolism in the brainstem, cerebellum, hippocampus and amygdala persisted over time and correlated with inflammation status.Conclusion Synergistic effects of systemic virus-mediated inflammation and transient hypoxia yield a dysfunction of the fronto-insular cortex, a signature of CNS involvement in neuro-COVID-19. This brain dysfunction is likely to be transient and almost reversible. The long-lasting brain hypermetabolism seems to reflect persistent inflammation processes