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We consider the mathematical analysis and numerical approximation of a system of nonlinear partial differential equations that arises in models that have relevance to steady isochoric flows of colloidal suspensions. The symmetric velocity gradient is assumed to be a monotone nonlinear function of the deviatoric part of the Cauchy stress tensor. We prove the existence of a unique weak solution to the problem, and under the additional assumption that the nonlinearity involved in the constitutive relation is Lipschitz continuous we also prove uniqueness of the weak solution. We then construct mixed finite element approximations of the system using both conforming and nonconforming finite element spaces. For both of these we prove the convergence of the method to the unique weak solution of the problem, and in the case of the conforming method we provide a bound on the error between the analytical solution and its finite element approximation in terms of the best approximation error from the finite element spaces. We propose first a Lions-Mercier type iterative method and next a classical fixed-point algorithm to solve the finite-dimensional problems resulting from the finite element discretisation of the system of nonlinear partial differential equations under consideration and present numerical experiments that illustrate the practical performance of the proposed numerical method., Comment: [v2] minor modifications; to appear in ESAIM: M2AN

We propose a novel algorithm for the approximation of surface-quasi geostrophic (SQG) flows modeled by a nonlinear partial differential equation coupling transport and fractional diffusion phenomena. The time discretization consists of an explicit strong-stability-preserving three-stage Runge-Kutta method while a flux-corrected-transport (FCT) method coupled with Dunford-Taylor representations of fractional operators is advocated for the space discretization. Standard continuous piecewise linear finite elements are employed and the algorithm does not have restrictions on the mesh structure nor on the computational domain. In the inviscid case, we show that the resulting scheme satisfies a discrete maximum principle property under a standard CFL condition and observe, in practice, its second-order accuracy in space. The algorithm successfully approximates several benchmarks with sharp transitions and fine structures typical of SQG flows. In addition, theoretical Kolmogorov energy decay rates are observed on a freely decaying atmospheric turbulence simulation.

We study the regularity of the solution to an obstacle problem for a class of integro–differential operators. The differential part is a second order elliptic operator, whereas the nonlocal part is given by the integral fractional Laplacian. The obtained smoothness is then used to design and analyze a finite element scheme.

A local discontinuous Galerkin (LDG) method for approximating large deformations of prestrained plates is introduced and tested on several insightful numerical examples in our previous computational work. This paper presents a numerical analysis of this LDG method, focusing on the free boundary case. The problem consists of minimizing a fourth order bending energy subject to a nonlinear and nonconvex metric constraint. The energy is discretized using LDG and a discrete gradient flow is used for computing discrete minimizers. We first show $\Gamma$-convergence of the discrete energy to the continuous one. Then we prove that the discrete gradient flow decreases the energy at each step and computes discrete minimizers with control of the metric constraint defect. We also present a numerical scheme for initialization of the gradient flow, and discuss the conditional stability of it., Comment: 29 pages, no figure

We propose a new nonconforming finite element algorithm to approximate the solution to the elliptic problem involving the fractional Laplacian. We first derive an integral representation of the bilinear form corresponding to the variational problem. The numerical approximation of the action of the corresponding stiffness matrix consists of three steps: (1) apply a sinc quadrature scheme to approximate the integral representation by a finite sum where each term involves the solution of an elliptic partial differential equation defined on the entire space, (2) truncate each elliptic problem to a bounded domain, (3) use the finite element method for the space approximation on each truncated domain. The consistency error analysis for the three steps is discussed together with the numerical implementation of the entire algorithm. The results of computations are given illustrating the error behavior in terms of the mesh size of the physical domain, the domain truncation parameter and the quadrature spacing parameter.

A reduced model for large deformations of prestrained plates consists of minimizing a second order bending energy subject to a nonconvex metric constraint. The former involves the second fundamental form of the middle plate and the latter is a restriction on its first fundamental form. We discuss a formal derivation of this reduced model along with an equivalent formulation that makes it amenable computationally. We propose a local discontinuous Galerkin (LDG) finite element approach that hinges on the notion of reconstructed Hessian. We design discrete gradient flows to minimize the ensuing nonconvex problem and to find a suitable initial deformation. We present several insightful numerical experiments, some of practical interest, and assess various computational aspects of the approximation process.

We present a computational model of thin elastic bilayers that undergo large bending isometric deformations when actuated by non-mechanical stimuli. We propose a discontinuous Galerkin approximation of the variational formulation discussed in Bartels et al. (2018). We showcase its advantages and good computational performance with configurations of interest in both engineering and medicine and either Dirichlet or free boundary conditions., V2 minor modifications

Partial differential equations posed on surfaces arise in a number of applications. In this survey we describe three popular finite element methods for approximating solutions to the Laplace–Beltrami problem posed on an n-dimensional surface γ embedded in R n + 1 : the parametric, trace, and narrow band methods. The parametric method entails constructing an approximating polyhedral surface Γ whose faces comprise the finite element triangulation. The finite element method is then posed over the approximate surface Γ in a manner very similar to standard FEM on Euclidean domains. In the trace method it is assumed that the given surface γ is embedded in an n + 1-dimensional domain Ω which has itself been triangulated. An n-dimensional approximate surface Γ is then constructed roughly speaking by interpolating γ over the triangulation of Ω, and the finite element space over Γ consists of the trace (restriction) of a standard finite element space on Ω to Γ. In the narrow band method the PDE posed on the surface is extended to a triangulated n + 1-dimensional band about γ whose width is proportional to the diameter of elements in the triangulation. In all cases we provide optimal a priori error estimates for the lowest order finite element methods, and we also present a posteriori error estimates for the parametric and trace methods. Our presentation focuses especially on the relationship between the regularity of the surface γ, which is never assumed better than of class C2, the manner in which γ is represented in theory and practice, and the properties of the resulting methods.

The usual approach to model reduction for parametric partial differential equations (PDEs) is to construct a linear space $V_n$ which approximates well the solution manifold $\mathcal{M}$ consisting of all solutions $u(y)$ with $y$ the vector of parameters. This linear reduced model $V_n$ is then used for various tasks such as building an online forward solver for the PDE or estimating parameters from data observations. It is well understood in other problems of numerical computation that nonlinear methods such as adaptive approximation, $n$-term approximation, and certain tree-based methods may provide improved numerical efficiency. For model reduction, a nonlinear method would replace the linear space $V_n$ by a nonlinear space $\Sigma_n$. This idea has already been suggested in recent papers on model reduction where the parameter domain is decomposed into a finite number of cells and a linear space of low dimension is assigned to each cell. Up to this point, little is known in terms of performance guarantees for such a nonlinear strategy. Moreover, most numerical experiments for nonlinear model reduction use a parameter dimension of only one or two. In this work, a step is made towards a more cohesive theory for nonlinear model reduction. Framing these methods in the general setting of library approximation allows us to give a first comparison of their performance with those of standard linear approximation for any general compact set. We then turn to the study these methods for solution manifolds of parametrized elliptic PDEs. We study a very specific example of library approximation where the parameter domain is split into a finite number $N$ of rectangular cells and where different reduced affine spaces of dimension $m$ are assigned to each cell. The performance of this nonlinear procedure is analyzed from the viewpoint of accuracy of approximation versus $m$ and $N$.

We construct a finite element approximation of a strain-limiting elastic model on a bounded open domain in $\mathbb{R}^d$, $d \in \{2,3\}$. The sequence of finite element approximations is shown to exhibit strong convergence to the unique weak solution of the model. Assuming that the material parameters featuring in the model are Lipschitz-continuous, and assuming that the weak solution has additional regularity, the sequence of finite element approximations is shown to converge with a rate. An iterative algorithm is constructed for the solution of the system of nonlinear algebraic equations that arises from the finite element approximation. An appealing feature of the iterative algorithm is that it decouples the monotone and linear elastic parts of the nonlinearity in the model. In particular, our choice of piecewise constant approximation for the stress tensor (and continuous piecewise linear approximation for the displacement) allows us to compute the monotone part of the nonlinearity by solving an algebraic system with $d(d+1)/2$ unknowns independently on each element in the subdivision of the computational domain. The theoretical results are illustrated by numerical experiments., Comment: [v3] modifications / simplifications in the proof of Theorem 7.1

Atazanavir (300 mg) boosted by ritonavir (100 mg) is the preferred third drug in pregnancy. However, there is still discordance on atazanavir dose increase during the third trimester.To evaluate plasma and intracellular atazanavir and ritonavir concentrations in HIV-infected women during pregnancy and after delivery.This was an observational study. HIV-infected pregnant patients treated with atazanavir/ritonavir plus either tenofovir/emtricitabine or abacavir/lamivudine had been prospectively enrolled after having signed a written informed consent form. Plasma and intracellular atazanavir and ritonavir Ctrough (24 ± 3 h after drug intake) were measured at each visit during the first, second and third trimesters and post-partum using validated HPLC-MS and HPLC-photodiode array methods (with direct evaluation of cellular volume). Data are described as median (IQR) and compared through non-parametric tests.Twenty-five patients were enrolled; at baseline, the median age was 32 years (27-35). All patients had plasma HIV RNA50 copies/mL; the median CD4+ count was 736 cells/mm3 (542-779). Atazanavir plasma concentrations were 441 ng/mL (261-1557), 710 ng/mL (338-1085), 556 ng/mL (334-1022) and 837 ng/mL (608-1757) during the first, second and third trimesters and post-partum, respectively; intracellular concentrations were 743 ng/mL (610-1928), 808 ng/mL (569-1620), 756 ng/mL (384-1074) and 706 ng/mL (467-2688), respectively. Atazanavir intracellular/plasma ratios were 1.32 (0.98-2.77), 1.34 (1.13-1.88), 1.38 (0.61-2.63) and 1.07 (0.56-2.69), respectively. Atazanavir intracellular concentrations and intracellular/plasma ratios showed non-significant changes over time (P 0.05).This is the first demonstration that intracellular atazanavir exposure remains unchanged during pregnancy supporting the standard 300/100 mg atazanavir/ritonavir dosing throughout pregnancy.

We study the numerical approximation of a time dependent equation involving fractional powers of an elliptic operator $L$ defined to be the unbounded operator associated with a Hermitian, coercive and bounded sesquilinear form on $H^1_0(\Omega)$. The time dependent solution $u(x,t)$ is represented as a Dunford Taylor integral along a contour in the complex plane. The contour integrals are approximated using sinc quadratures. In the case of homogeneous right-hand-sides and initial value $v$, the approximation results in a linear combination of functions $(z_qI-L)^{-1}v\in H^1_0(\Omega)$ for a finite number of quadrature points $z_q$ lying along the contour. In turn, these quantities are approximated using complex valued continuous piecewise linear finite elements. Our main result provides $L^2(\Omega)$ error estimates between the solution $u(\cdot,t)$ and its final approximation. Numerical results illustrating the behavior of the algorithms are provided., Comment: 20 pages, 4 figures

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 $H({\rm div})$-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 $H^1$ conformity of the velocity field. In addition our method exactly satisfies the incompressibility constraint in the surface Stokes problem. Secondly, 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., 29 pages

The use of combination anti-retroviral therapy (cART) correlates with longer and healthier life and with nearly normal life expectancy in people living with HIV. However, cART does not completely restore health. Chronic immune activation and inflammation persist in treated patients and have been described as predictors for clinical events and mortality in HIV-infected patients. Limited information is available on the impact of the various cART regimens on inflammation/immunoactivation. The aim of this work was to explore the impact of elvitegravir, dolutegravir, raltegravir (integrase strand transfer inhibitors, INSTIs) and atazanavir (protease inhibitor, PI) on several soluble markers of immune activation and inflammation during the first year of effective combination anti-retroviral therapy (cART). We conducted an observational retrospective cohort study in HIV-infected cART-naive patients who initiated an INSTI or atazanavir regimen between March 2015 and February 2016 and a serum sample was available at baseline, 6 and 12 months after initiation. We compared the trend of D-Dimer, TNF- α, IL-2, IL-6, IL-7, IL-10, CCL4/MIP1-β, CCL5/RANTES, s-CD14, s-CD163, hs-CRP levels among the 4 arms of treatment. Percentage of variation from baseline was also measured for all markers. A total of 36 patients were included. We observed heterogeneous modifications in inflammation markers among arms. In particular, we noted that EVG have significant negative effect on s-CD14, hs-CRP, IL-6 and D-Dimer in respect to other INSTIs and this different effect occurs mainly during the first 6 months of cART. IL-7 values increased in the three arms with INSTIs (significantly only in EGV, 159.8%, p = 0.0003) and decreased significantly in patients on PI (−48.96%; p = 0.04) over the period. In conclusion, our results provide further data on changes of inflammatory marker levels, especially for the new INSTIs. Our data show that among INSTIs, EVG seems to have a worse impact on inflammation.

We consider the numerical approximation of the spectral fractional diffusion problem based on the so called Balakrishnan representation. The latter consists of an improper integral approximated via quadratures. At each quadrature point, a reaction–diffusion problem must be approximated and is the method bottle neck. In this work, we propose to reduce the computational cost using a reduced basis strategy allowing for a fast evaluation of the reaction–diffusion problems. The reduced basis does not depend on the fractional power s for 0 < smin ⩽ s ⩽ smax < 1. It is built offline once for all and used online irrespectively of the fractional power. We analyze the reduced basis strategy and show its exponential convergence. The analytical results are illustrated with insightful numerical experiments.

Motivated by numerical methods for solving parametric partial differential equations, this paper studies the approximation of multivariate analytic functions by algebraic polynomials. We introduce various anisotropic model classes based on Taylor expansions, and study their approximation by finite dimensional polynomial spaces $${{\mathcal {P}}}_\Lambda $$ described by lower sets $$\Lambda $$ . Given a budget n for the dimension of $${{\mathcal {P}}}_\Lambda $$ , we prove that certain lower sets $$\Lambda _n$$ , with cardinality n, provide a certifiable approximation error that is in a certain sense optimal, and that these lower sets have a simple definition in terms of simplices. Our main goal is to obtain approximation results when the number of variables d is large and even infinite, and so we concentrate almost exclusively on the case $$d=\infty $$ . We also emphasize obtaining results which hold for the full range $$n\ge 1$$ , rather than asymptotic results that only hold for n sufficiently large. In applications, one typically wants n small to comply with computational budgets.

We propose a new discontinuous Galerkin (dG) method for a geometrically nonlinear Kirchhoff plate model for large isometric bending deformations. The minimization problem is nonconvex due to the isometry constraint. We present a practical discrete gradient flow that decreases the energy and computes discrete minimizers that satisfy a prescribed discrete isometry defect. We prove $\Gamma$-convergence of the discrete energies and discrete global minimizers. We document the flexibility and accuracy of the dG method with several numerical experiments., Comment: 38 pages

We present a new AFEM for the Laplace-Beltrami operator with arbitrary polynomial degree on parametric surfaces, which are globally $W^1_\infty$ and piecewise in a suitable Besov class embedded in $C^{1,\alpha}$ with $\alpha \in (0,1]$. The idea is to have the surface sufficiently well resolved in $W^1_\infty$ relative to the current resolution of the PDE in $H^1$. This gives rise to a conditional contraction property of the PDE module. We present a suitable approximation class and discuss its relation to Besov regularity of the surface, solution, and forcing. We prove optimal convergence rates for AFEM which are dictated by the worst decay rate of the surface error in $W^1_\infty$ and PDE error in $H^1$., Comment: 51 pages, the published version contains an additional glossary

The bending of bilayer plates is a mechanism that allows for large deformations via small externally induced lattice mismatches of the underlying materials. Its mathematical modeling, discussed herein, consists of a nonlinear fourth-order problem with a pointwise isometry constraint. A discretization based on Kirchhoff quadrilaterals is devised and its Γ-convergence is proved. An iterative method that decreases the energy is proposed, and its convergence to stationary configurations is investigated. Its performance, as well as reduced model capabilities, are explored via several insightful numerical experiments involving large (geometrically nonlinear) deformations.© 2015 Wiley Periodicals, Inc.

We propose a finite element method for the numerical simulation of electroconvection of thin liquid crystals. The liquid is located in between two concentric circular electrodes which are either assumed to be of infinite height or slim. Each configuration results in a different nonlocal electro-magnetic model defined on a two dimensional bounded domain. The numerical method consists in approximating the surface charge density, the liquid velocity and pressure, and the electric potential in the two dimensional liquid region. Finite elements for the space discretization coupled with standard time stepping methods are put forward. Unlike for the infinite electrodes configuration, our numerical simulations indicate that slim electrodes are favorable for electroconvection to occur and are able to sustain the phenomena over long period of time. Furthermore, we provide a numerical study on the influence of the three main parameters of the system: the Rayleigh number, the Prandtl number and the electrodes aspect ratio., Comment: 28 pages

A stochastic model corresponding to a simplified Hookean dumbbells viscoelastic fluid is considered, the convective terms being disregarded. Existence on a fixed time interval is proved provided the data are small enough, using the implicit function theorem and a maximum regularity property for a three fields Stokes problem

We prove new a posteriori error estimates for surface finite element methods (SFEM). Surface FEM approximate solutions to PDE posed on surfaces. Prototypical examples are elliptic PDE involving the Laplace-Beltrami operator. Typically the surface is approximated by a polyhedral or higher-order polynomial approximation. The resulting FEM exhibits both a geometric consistency error due to the surface approximation and a standard Galerkin error. A posteriori estimates for SFEM 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 SFEM. However, previous a posteriori estimates using general parametric surface representations are suboptimal by one order on $C^2$ 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 {\it practical} access only to a general parametric representation of the surface, but using the distance function as a {\it 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.

Atazanavir-ritonavir concentrations change over time during pregnancy in HIV-positive patients; the impact of genetic variants is unknown. Twenty patients were enrolled in this study; plasma and intracellular concentrations of antiretrovirals were measured, in addition to single-nucleotide polymorphisms in transport-related genes. Linear logistic regression showed that genetic variants in organic-anion-transporter-1B1- and pregnane-X-receptor-encoding genes affected third-trimester atazanavir exposure. In this prospective study, genetic variants partially explained the observed interpatient variability in third-trimester exposure to antiretrovirals.

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.

We consider a fictitious domain formulation of an elliptic partial differential equation and approximate the resulting saddle-point system using a nested inexact preconditioned Uzawa iterative algorithm, which consists of three nested loops. In the outer loop the trial space for the Galerkin approximation of the Lagrange multiplier is enlarged. The intermediate loop solves this Galerkin system by a damped preconditioned Richardson iteration. Each iteration of the latter involves solving an elliptic problem on the fictitious domain whose solution is approximated by an adaptive finite element method in the inner loop. We prove that the overall method converges with the best possible rate and illustrate numerically our theoretical findings.

Bouncing jets are fascinating phenomenons occurring under certain conditions when a jet impinges on a free surface. This effect is observed when the fluid is Newtonian and the jet falls in a bath undergoing a solid motion. It occurs also for non-Newtonian fluids when the jets falls in a vessel at rest containing the same fluid. We investigate numerically the impact of the experimental setting and the rheological properties of the fluid on the onset of the bouncing phenomenon. Our investigations show that the occurrence of a thin lubricating layer of air separating the jet and the rest of the liquid is a key factor for the bouncing of the jet to happen. The numerical technique that is used consists of a projection method for the Navier-Stokes system coupled with a level set formulation for the representation of the interface. The space approximation is done with adaptive finite elements. Adaptive refinement is shown to be very important to capture the thin layer of air that is responsible for the bouncing.

We present and study a novel numerical algorithm to approximate the action of $T^\beta:=L^{-\beta}$ where $L$ is a symmetric and positive definite unbounded operator on a Hilbert space $H_0$. The numerical method is based on a representation formula for $T^{-\beta}$ in terms of Bochner integrals involving $(I+t^2L)^{-1}$ for $t\in(0,\infty)$. To develop an approximation to $T^\beta$, we introduce a finite element approximation $L_h$ to $L$ and base our approximation to $T^\beta$ on $T_h^\beta:= L_h^{-\beta}$. The direct evaluation of $T_h^{\beta}$ is extremely expensive as it involves expansion in the basis of eigenfunctions for $L_h$. The above mentioned representation formula holds for $T_h^{-\beta}$ and we propose three quadrature approximations denoted generically by $Q_h^\beta$. The two results of this paper bound the errors in the $H_0$ inner product of $T^\beta-T_h^\beta\pi_h$ and $T_h^\beta-Q_h^\beta$ where $\pi_h$ is the $H_0$ orthogonal projection into the finite element space. We note that the evaluation of $Q_h^\beta$ involves application of $(I+(t_i)^2L_h)^{-1}$ with $t_i$ 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 $T_h^\beta-Q_h^\beta$ and the finite element error $T^\beta-T_h^\beta\pi_h$.

Discontinuity in viscosities is of interest in many applications. Classical adaptive numerical methods perform under the restricting assumption that the discontinuities of the viscosity are captured by the initial partition. This excludes applications where the jump of the viscosity takes place across curves, manifolds or at a priori unknown positions. We present a novel estimate measuring the distortion of the viscosity in Lq for a q 2. We show that the latter holds on any bounded Lipschitz domain provided the data belongs to a smaller class than those required to obtain well-posedness. Based on this theory, we introduce adaptive finite element methods which approximate the solution of Stokes equations with possible discontinuous viscosities. We prove that these algorithms are quasi-optimal in terms of error compared to the number of cells. Finally, the performance of the adaptive algorithm is numerically illustrated on insightful examples.

With the development of combination antiretroviral therapy (cART), the first generation of perinatally HIV-infected children has reached young adulthood. A retrospective study was conducted on perinatally HIV-infected young adults after transition to adult care in Brescia (Northern Italy). Twenty-four patients were transferred to Infectious Disease outpatient Clinic from Pediatric Clinic between 2004 and 2016. Median age at transition was 18 years. 37.5% were male, and 75% were Italian. Median CD4+ T-cell count was 534 cell/μL, and 9/24 presented detectable HIV-RNA at the time of transition. At month 12 after transition, median CD4+ T-cell count was 626 cell/μL, and HIV-RNA was still detectable in 25% of patients. Nineteen patients were still in care at the end of follow-up (median of 52 months); 100% on cART, with undetectable HIV-RNA and a median CD4+ T-cell count of 716 cell/μL. After transition, cART regimen was modified in 14/19 patients (in 13 of them it was modified at least twice). Resist...

We consider the finite element approximation of fractional powers of regularly accretive operators via the Dunford-Taylor integral approach. We use a sinc quadrature scheme to approximate the Balakrishnan representation of the negative powers of the operator as well as its finite element approximation. We improve the exponentially convergent error estimates from [A. Bonito, J. E. Pasciak, IMA J. Numer. Anal. (2016) 00, 1-29] by reducing the regularity required on the data. Numerical experiments illustrating the new theory are provided., 3 figures

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 $E(t)$ for the initial value problem can be written as a Dunford-Taylor integral involving the Mittag-Leffler function $e_{\alpha,1}$ and the resolvent of the underlying (non-fractional) spatial operator over an appropriate integration path in the complex plane. Here $\alpha$ denotes the order of the fractional time derivative. The solution for the non-homogeneous problem can be written as a convolution involving an operator $W(t)$ and the forcing function $F(t)$. 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 $1/h$. The fully discrete method for the initial value problem is developed from the semi-discrete approximation by applying an exponentially convergent sinc quadrature technique to approximate the Dunford-Taylor integral of the discrete operator and is free of any time stepping. 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 $W_h(s)$, (the semi-discrete approximation to $W(s)$) 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., Comment: 29 pages, 5 figures

We present a mathematical model of polymer bilayers that undergo large bending deformations when actuated by non-mechanical stimuli such as thermal effects. The simple model captures a large class of nonlinear bending effects and can be discretized with standard plate elements. We devise a fully practical iterative scheme and apply it to the simulation of folding of several practically useful compliant structures comprising of thin elastic layers., 27 pages

In this article we investigate, via numerical computations, the intersection properties of the nodal set of the eigenfunctions of the Laplace-Beltrami operator for smooth surfaces in $R^3$ (the nodal set of a continuous function is the set of those points at which the function vanishes). First, we briefly discuss the numerical solution of the eigenvalue/eigenfunction problem for the Laplace-Beltrami operator on bounded surfaces of $R^3$, and then consider some specific surfaces and visualize how the nodal lines intersect (or not) depending of the symmetries verified by the surface. After validating our computational methodology with the surface of a ring torus, we will investigate a simple surface without symmetry and observe that in that case the nodal set of the computed eigenfunctions consists of non intersecting lines, suggesting some conjecture. We observe also that for the above symmetry-free surface, the number of connected components of the nodal set varies non-monotonically with the rank of the associated eigenvalue (assuming that the eigenvalues are ordered by increasing value).

We establish the L 2 L^2 -stability of an entropy viscosity technique applied to nonlinear scalar conservation equations. First- and second-order explicit time-stepping techniques using continuous finite elements in space are considered. The method is shown to be stable independently of the polynomial degree of the space approximation under the standard CFL condition.

We derive optimal a priori error estimates for discontinuous Galerkin (dG) time discrete schemes of any order applied to an advection---diffusion model defined on moving domains and written in the Arbitrary Lagrangian Eulerian (ALE) framework. Our estimates hold without any restrictions on the time steps for dG with exact integration or Reynolds' quadrature. They involve a mild restriction on the time steps for the practical Runge---Kutta---Radau methods of any order. The key ingredients are the stability results shown earlier in Bonito et al. (Time-discrete higher order ALE formulations: stability, 2013) along with a novel ALE projection. Numerical experiments illustrate and complement our theoretical results.

Arbitrary Lagrangian Eulerian (ALE) formulations deal with PDEs on deformable domains upon extending the domain velocity from the boundary into the bulk with the purpose of keeping mesh regularity. This arbitrary extension has no effect on the stability of the PDE but may influence that of a discrete scheme. We examine this critical issue for higher-order time stepping without space discretization. We propose time-discrete discontinuous Galerkin (dG) numerical schemes of any order for a time-dependent advection-diffusion-model problem in moving domains, and study their stability properties. The analysis hinges on the validity of the Reynolds' identity for dG. Exploiting the variational structure and assuming exact integration, we prove that our conservative and nonconservative dG schemes are equivalent and unconditionally stable. The same results remain true for piecewise polynomial ALE maps of any degree and suitable quadrature that guarantees the validity of the Reynolds' identity. This approach general...

This paper considers the Dirichlet problem $$ -\mathrm{div}(a\nabla u_a)=f \quad \hbox{on}\,\,\ D, \qquad u_a=0\quad \hbox{on}\,\,\partial D, $$ for a Lipschitz domain $D\subset \mathbb R^d$, where $a$ is a scalar diffusion function. For a fixed $f$, we discuss under which conditions is $a$ uniquely determined and when can $a$ be stably recovered from the knowledge of $u_a$. A first result is that whenever $a\in H^1(D)$, with $0, Comment: 25 pages

We investigate the benefits of operator splitting methods in the context of computational fluid dynamics. In particular, we exploit their capacity at handling free surface flows and a large variety of physical phenomena in a flexible way. A mathematical and computational framework is presented for the numerical simulation of free surface flows, where the operator splitting strategy allows to separate inertial effects from the other effects. The method of characteristics on a fine structured grid is put forward to accurately approximate the inertial effects while continuous piecewise polynomial finite element associated with a coarser subdivision made of simplices is advocated for the other effects. In addition, the splitting strategy also allows modularity, and in a straightforward manner rheological model change for the fluid. We will emphasize this flexibility by treating Newtonian flows, visco-elastic flows, multi-phase, and multi-density immiscible incompressible Newtonian flows. The numerical framework is thoroughly presented; the test case of the filling of a cylindrical tube with potential die swell in an extrusion process is taken as the main illustration of the advantages of operator splitting.

Neurocognitive disorders are emerging, probably underestimated, complications in HIV-infected people. The aim of the study was to assess neurocognitive profiles of newly detected HIV-infected patients. We performed an observational retrospective single-cohort study. Illiterates and patients with neurologic symptoms or previous psychiatric diagnosis were excluded. Neuropsychological profiles were assessed using a validated battery of neuropsychological tests. We included 206 patients; with males representing the majority of them (85%). Risk factors for HIV acquisition were unprotected sexual intercourse (homo/bisexual in 39.8% and heterosexual in 60.2%). Thirty-nine patients (18.9%) were previous injection drug users, while 41 (19.9%) were alcohol abusers. Mean education was 11.1 years (SD--standard deviation--3.7). A high prevalence of HIV-associated neurocognitive disorders (HAND, 47.1%) was present in HIV-infected patients: particularly, asymptomatic neurocognitive impairment (ANI) was found in 30.6%, mild neurocognitive disorder (MND) in 15% and HIV-associated dementia (HAD) in 1.5%. Male gender, low degree of education, AIDS diagnosis and gepatitis B virus (HBV) co-infection were factors independently associated with HAND in a multivariable logistic regression model. Our data suggest that patient-specific factors and AIDS diagnosis have a certain kind of impact in HAND occurrence. A complete neuropsychological screening must be recommended in all patients at HIV-infection diagnosis.

We design and analyze variational and non-variational multigrid algorithms for the Laplace-Beltrami operator on a smooth and closed surface. In both cases, a uniform convergence for the V -cycle algorithm is obtained provided the surface geometry is captured well enough by the coarsest grid. The main argument hinges on a perturbation analysis from an auxiliary variational algorithm defined directly on the smooth surface. In addition, the vanishing mean value constraint is imposed on each level, thereby avoiding singular quadratic forms without adding additional computational cost. Numerical results supporting our analysis are reported. In particular, the algorithms perform well even when applied to surfaces with a large aspect ratio.