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We present and study a novel numerical algorithm to approximate the action of Tβ:=L−βT^\\beta :=L^{-\\beta } where LL is a symmetric and positive definite unbounded operator on a Hilbert space H0H_0. The numerical method is based on a representation formula for T−βT^{-\\beta } in terms of Bochner integrals involving (I+t2L)−1(I+t^2L)^{-1} for t∈(0,∞)t\\in (0,\\infty ). To develop an approximation to TβT^\\beta, we introduce a finite element approximation LhL_h to LL and base our approximation to TβT^\\beta on Thβ:=Lh−βT_h^\\beta := L_h^{-\\beta }. The direct evaluation of ThβT_h^{\\beta } is extremely expensive as it involves expansion in the basis of eigenfunctions for LhL_h. The above mentioned representation formula holds for Th−βT_h^{-\\beta } and we propose three quadrature approximations denoted generically by QhβQ_h^\\beta. The two results of this paper bound the errors in the H0H_0 inner product of Tβ−ThβπhT^\\beta -T_h^\\beta \\pi _h and Thβ−QhβT_h^\\beta -Q_h^\\beta where πh\\pi _h is the H0H_0 orthogonal projection into the finite element space. We note that the evaluation of QhβQ_h^\\beta involves application of (I+(ti)2Lh)−1(I+(t_i)^2L_h)^{-1} with tit_i being either a quadrature point or its inverse. Efficient solution algorithms for these problems are available and the problems at different quadrature points can be straightforwardly solved in parallel. Numerical experiments illustrating the theoretical estimates are provided for both the quadrature error Thβ−QhβT_h^\\beta -Q_h^\\beta and the finite element error Tβ−ThβπhT^\\beta -T_h^\\beta \\pi _h.

In this paper, we develop a numerical scheme for the space-time fractional parabolic equation, i.e. an equation involving a fractional time derivative and a fractional spatial operator. Both the initial value problem and the non-homogeneous forcing problem (with zero initial data) are considered. The solution operator for the initial value problem can be written as a Dunford–Taylor integral involving the Mittag-Leffler function and the resolvent of the underlying (non-fractional) spatial operator over an appropriate integration path in the complex plane. Here α denotes the order of the fractional time derivative. The solution for the non-homogeneous problem can be written as a convolution involving an operator and the forcing function . We develop and analyze semi-discrete methods based on finite element approximation to the underlying (non-fractional) spatial operator in terms of analogous Dunford–Taylor integrals applied to the discrete operator. The space error is of optimal order up to a logarithm of . The fully discrete method for the initial value problem is developed from the semi-discrete approximation by applying a sinc quadrature technique to approximate the Dunford–Taylor integral of the discrete operator and is free of any time stepping. The sinc quadrature of step size involves nodes and results in an additional error. To approximate the convolution appearing in the semi-discrete approximation to the non-homogeneous problem, we apply a pseudo-midpoint quadrature. This involves the average of , (the semi-discrete approximation to ) over the quadrature interval. This average can also be written as a Dunford–Taylor integral. We first analyze the error between this quadrature and the semi-discrete approximation. To develop a fully discrete method, we then introduce sinc quadrature approximations to the Dunford–Taylor integrals for computing the averages. We show that for a refined grid in time with a mesh of intervals, the error between the semi-discrete and fully discrete approximation is . We also report the results of numerical experiments that are in agreement with the theoretical error estimates.

Pyelo-hepatic abscess is a rare complication of upper urinary tract infections (UTIs). We describe a case of polymicrobial pyelo-hepatic abscess in an immunocompetent patient. A 71-year-old male patient with a double-J stent for right ureteral lithiasis was admitted in our Infectious Diseases Department for a pyelo-hepatic abscess. Despite a targeted antibiotic therapy against an extended spectrum betalactamase-negative Escherichia coli, the patient did not improve. Further examinations revealed a possible polymicrobial aetiology, including Candida spp. and E. coli resistant to piperacillin/tazobactam but sensitive to third-generation cephalosporins. To date, a paucity of articles regarding pyelo-hepatic abscess exist, consisting mostly of case reports. Urinary stones and a ureteral stent indwelling time exceeding 90 days are known risk factors for upper UTIs and for bacterial dissemination in contiguous organs. Pyelo-hepatic abscesses usually involve Gram-negative bacilli, but they can be polymicrobial, including fungi. As a range of factors could limit the efficacy of antibiotics inside an encapsulated lesion and might contribute to the selection of resistant species during treatment, clinicians should be aware of this complication and try to prevent this event by acting on the main modifiable risk factor.

The Stokes equation posed on surfaces is important in some physical models, but its numerical solution poses several challenges not encountered in the corresponding Euclidean setting. These include the fact that the velocity vector should be tangent to the given surface and the possible presence of degenerate modes (Killing fields) in the solution. We analyze a surface finite element method which provides solutions to these challenges. We consider an interior penalty method based on the well-known Brezzi-Douglas-Marini H(div)-conforming finite element space. The resulting spaces are tangential to the surface but require penalization of jumps across element interfaces in order to weakly maintain H¹ conformity of the velocity field. In addition our method exactly satisfies the incompressibility constraint in the surface Stokes problem. Second, we give a method which robustly filters Killing fields out of the solution. This problem is complicated by the fact that the dimension of the space of Killing fields may change with small perturbations of the surface. We first approximate the Killing fields via a Stokes eigenvalue problem and then give a method which is asymptotically guaranteed to correctly exclude them from the solution. The properties of our method are rigorously established via an error analysis and illustrated via numerical experiments.

Proofs of convergence of adaptive finite element methods (AFEMs) for approximating eigenvalues and eigenfunctions of linear elliptic problems have been given in several recent papers. A key step in establishing such results for multiple and clustered eigenvalues was provided by Dai, He, and Zhou in [IMA J. Numer. Anal., 35 (2015), pp. 1934-1977], who proved convergence and optimality of AFEMs for eigenvalues of multiplicity greater than one. There it was shown that a theoretical (noncomputable) error estimator for which standard convergence proofs apply is equivalent to a standard computable estimator on sufficiently fine grids. In [Numer. Math., 130 (2015), pp. 467-496], Gallistl used a similar tool to prove that a standard AFEM for controlling eigenvalue clusters for the Laplacian using continuous piecewise linear finite element spaces converges with optimal rate. When considering either higher-order finite element spaces or nonconstant diffusion coefficients, however, the arguments of Dai, He, and Zhou and Gallistl do not yield equivalence of the practical and theoretical estimators for clustered eigenvalues. In this article we provide this missing key step, thus showing that standard adaptive FEMs for clustered eigenvalues employing elements of arbitrary polynomial degree converge with optimal rate. We additionally establish that a key user-defined input parameter in the AFEM, the bulk marking parameter, may be chosen entirely independently of the properties of the target eigenvalue cluster. All of these results assume a fineness condition on the initial mesh in order to ensure that the nonlinearity is sufficiently resolved.

We prove new a posteriori error estimates for surface finite element methods (SFEMs). SFEMs approximate solutions to PDEs posed on surfaces. Prototypical examples are elliptic PDEs involving the Laplace-Beltrami operator. Typically the surface is approximated by a polyhedral or higher-order polynomial approximation. The resulting finite element method exhibits both a geometric consistency error due to the surface approximation and a standard Galerkin error. A posteriori estimates for SFEMs require practical access to geometric information about the surface in order to computably bound the geometric error. It is thus advantageous to allow for maximum flexibility in representing surfaces in practical codes when proving a posteriori error estimates for SFEMs. However, previous a posteriori estimates using general parametric surface representations are suboptimal by one order on C² surfaces. Proofs of error estimates optimally reflecting the geometric error instead employ the closest point projection, which is defined using the signed distance function. Because the closest point projection is often unavailable or inconvenient to use computationally, a posteriori estimates using the signed distance function have notable practical limitations. We merge these two perspectives by assuming practical access only to a general parametric representation of the surface, but using the distance function as a theoretical tool. This allows us to derive sharper geometric estimators which exhibit improved experimentally observed decay rates when implemented in adaptive surface finite element algorithms.

The Babuška or plate paradox concerns the failure of convergence when a domain with curved boundary is approximated by polygonal domains in linear bending problems with simply supported boundary conditions. It can be explained via a boundary integral representation of the total Gaussian curvature that is part of the Kirchhoff–Love bending energy. It is shown that the paradox also occurs for a nonlinear bending-folding model which enforces vanishing Gaussian curvature. A simple remedy that is compatible with simplicial finite-element methods to avoid incorrect convergence is devised.

Elliptic partial differential equations on surfaces play an essential role in geometry, relativity theory, phase transitions, materials science, image processing, and other applications. They are typically governed by the Laplace-Beltrami operator. We present and analyze approximations by surface finite element methods (SFEM) of the Laplace-Beltrami eigenvalue problem. As for SFEM for source problems, spectral approximation is challenged by two sources of errors: the geometric consistency error due to the approximation of the surface and the Galerkin error corresponding to finite element resolution of eigenfunctions. We show that these two error sources interact for eigenfunction approximations as for the source problem. The situation is different for eigenvalues, where a novel situation occurs for the geometric consistency error: The degree of the geometric error depends on the choice of interpolation points used to construct the approximate surface. Thus the geometric consistency term can sometimes be made to converge faster than in the eigenfunction case through a judicious choice of interpolation points.

Terrestrial ecosystems represent a major sink for atmospheric carbon (C) and temperate forests play an important role in global C cycling, contributing to lower atmospheric carbon dioxide (CO 2 ) concentration through photosynthesis. The Intergovernmental Panel of Climate Change highlights that the forestry sector has great potential to decrease atmospheric CO 2 concentration compared to other sectoral mitigation activities. The aim of this study was to evaluate CO 2 sequestration (CO 2 S) capability of Fagus sylvatica (beech) growing in the Orfento Valley within Majella National Park (Abruzzo, Italy). We compared F. sylvatica areas subjected to thinning (one high-forest and one coppice) and no-management areas (two high-forests and two coppices). The results show a mean CO 2 S of 44.3 ± 2.6 Mg CO 2  ha −1  a −1 , corresponding to 12.1 ± 0.7 Mg C ha −1  a −1 the no-managed areas having a 28% higher value than the managed areas. The results highlight that thinning that allows seed regeneration can support traditional management practices such as civic use in some areas while no management should be carried out in the reserve in order to give priority to the objective of conservation and naturalistic improvement of the forest heritage.