arXiv subjects

Andrea Bonito

Publications and source records attributed to Andrea Bonito.

49 records · Page 3Linked to original sources

arXiv2017

Modeling and simulation of thermally actuated bilayer plates

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.

math.NA↗

arXiv2017

Numerical Approximation of Space-time Fractional Parabolic Equations

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 $e_{α,1}$ 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 $W(t)$ 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.

math.NA↗

arXiv2017

Numerical Methods for Fractional Diffusion

We present three schemes for the numerical approximation of fractional diffusion, which build on different definitions of such a non-local process. The first method is a PDE approach that applies to the spectral definition and exploits the extension to one higher dimension. The second method is the integral formulation and deals with singular non-integrable kernels. The third method is a discretization of the Dunford-Taylor formula. We discuss pros and cons of each method, error estimates, and document their performance with a few numerical experiments.

math.NA↗

arXiv2016

Diffusion Coefficients Estimation for Elliptic Partial Differential Equations

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 $0<λ\le a\le Λ$ on $D$, and $f\in L_\infty(D)$ is strictly positive, then $$ \|a-b\|_{L_2(D)}\le C\|u_a-u_b\|_{H_0^1(D)}^{1/6}. $$ More generally, it is shown that the assumption $a\in H^1(D)$ can be weakened to $a\in H^s(D)$, for certain $s<1$, at the expense of lowering the exponent $1/6$ to a value that depends on $s$.

math.AP↗

arXiv2016

High-Order AFEM for the Laplace-Beltrami Operator: Convergence Rates

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 $C^{1,α}$ with $α\in (0,1]$. 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 $H^1$.

math.NA↗

arXiv2016

The Approximation of Parabolic Equations Involving Fractional Powers of Elliptic Operators

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 $H^1_0(Ω)$. 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 $(z_qI-L)^{-1}v\in H^1_0(Ω)$ 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 $L^2(Ω)$ error estimates between the solution $u(\cdot,t)$ and its final approximation. Numerical results illustrating the behavior of the algorithms are provided.

math.NA↗

arXiv2016

Numerical Approximation of Fractional Powers of Regularly Accretive Operators

We study the numerical approximation of fractional powers of accretive operators in this paper. Namely, if $A$ is the accretive operator associated with an accretive sesquilinear form $A(\cdot,\cdot)$ defined on a Hilbert space $\mathbb V$ contained in $L^2(Ω)$, we approximate $A^{-β}$ for $β\in (0,1)$. The fractional powers are defined in terms of the so-called Balakrishnan integral formula. Given a finite element approximation space $\mathbb V_h\subset \mathbb V$, $A^{-β}$ is approximated by $A_h^{-β}π_h$ where $A_h$ is the operator associated with the form $A(\cdot,\cdot)$ restricted to $\mathbb V_h$ and $π_h$ is the $L^2(Ω)$-projection onto $\mathbb V_h$. We first provide error estimates for $(A^β-A_h^βπ_h)f$ in Sobolev norms with index in [0,1] for appropriate $f$. These results depend on elliptic regularity properties of variational solutions involving the form $A(\cdot,\cdot)$ and are valid for the case of less than full elliptic regularity. We also construct and analyze an exponentially convergent sinc quadrature approximation to the Balakrishnan integral defining $A_h^βπ_h f$. Finally, the results of numerical computations illustrating the proposed method are given.

math.NA↗

arXiv2016

Convergence and optimality of higher-order adaptive finite element methods for eigenvalue clusters

Proofs of convergence of adaptive finite element methods for the approximation of eigenvalues and eigenfunctions of linear elliptic problems have been given in a several recent papers. A key step in establishing such results for multiple and clustered eigenvalues was provided by Dai et. al. (2014), who proved convergence and optimality of AFEM for eigenvalues of multiplicity greater than one. There it was shown that a theoretical (non-computable) error estimator for which standard convergence proofs apply is equivalent to a standard computable estimator on sufficiently fine grids. Gallistl (2015) used a similar tool in order to prove that a standard adaptive FEM 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 non-constant diffusion coefficients, however, the arguments of Dai et. al. and Gallistl do not yield equivalence of the practical and theoretical estimators for clustered eigenvalues. In this note we provide this missing key step, thus showing that standard adaptive FEM 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.

math.NA↗

arXiv2015

Bilayer Plates: Model Reduction, $Γ$-Convergent Finite Element Approximation and Discrete Gradient Flow

The bending of bilayer plates is a mechanism which 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.

math.NA↗

arXiv2015

An Interior Penalty Method with $C^0$ Finite Elements for the Approximation of the Maxwell Equations in Heterogeneous Media: Convergence Analysis with Minimal Regularity

The present paper proposes and analyzes an interior penalty technique using $C^0$-finite elements to solve the Maxwell equations in domains with heterogeneous properties. The convergence analysis for the boundary value problem and the eigenvalue problem is done assuming only minimal regularity in Lipschitz domains. The method is shown to converge for any polynomial degrees and to be spectrally correct.

math.NA↗

arXiv2015

Numerical Simulations of Bouncing Jets

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.

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arXiv2013

Numerical Approximation of Fractional Powers of Elliptic Operators

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

math.NA↗

arXiv2013

Adaptive Finite Element Methods for Elliptic Problems with Discontinuous Coefficients

Elliptic partial differential equations (PDEs) with discontinuous diffusion coefficients occur in application domains such as diffusions through porous media, electro-magnetic field propagation on heterogeneous media, and diffusion processes on rough surfaces. The standard approach to numerically treating such problems using finite element methods is to assume that the discontinuities lie on the boundaries of the cells in the initial triangulation. However, this does not match applications where discontinuities occur on curves, surfaces, or manifolds, and could even be unknown beforehand. One of the obstacles to treating such discontinuity problems is that the usual perturbation theory for elliptic PDEs assumes bounds for the distortion of the coefficients in the $L_\infty$ norm and this in turn requires that the discontinuities are matched exactly when the coefficients are approximated. We present a new approach based on distortion of the coefficients in an $L_q$ norm with $q<\infty$ which therefore does not require the exact matching of the discontinuities. We then use this new distortion theory to formulate new adaptive finite element methods (AFEMs) for such discontinuity problems. We show that such AFEMs are optimal in the sense of distortion versus number of computations, and report insightful numerical results supporting our analysis.

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