A multigrid method with unstructured adaptive grids for steady Euler equations

The multigrid method based on multi-stage Jacobi relaxation, earlier developed by the authors for structured grid calculations with Euler equations, is extended to unstructured grid applications. The meshes are generated with Delaunay triangulation algorithms and are adapted to the flow solution.     

A POSTERIORI ERROR ESTIMATES FOR FINITE ELEMENT APPROXIMATIONS OF THE CAHN-HILLIARD EQUATION AND THE HELE-SHAW FLOW

This paper develops a posteriori error estimates of residual type for conforming and mixed finite element approximations of the fourth order Cahn-Hilliard equation ut ＋ △（ε△Au-ε^-1f（u）） = 0. It is shown that the a posteriori error bounds depends on ε^-1 only in some low polynomial order, instead of exponential order. Using these a posteriori error estimates, we construct at2 adaptive algorithm for computing the solution of the Cahn- Hilliard equation and its sharp interface limit, the Hele-Shaw flow. Numerical experiments are presented to show the robustness and effectiveness of the new error estimators and the proposed adaptive algorithm.     

A wavelet adaptive Newton method for the solution of nonlinear equations

For the solution of nonlinear equations, we present an adaptive wavelet scheme, which couples an inexact Newton method and the idea of nonlinear wavelet approximation. In particular, we obtain a result of quadratic convergence.     

Simple Monte Carlo and the Metropolis algorithm

We study the integration of functions with respect to an unknown density. Information is available as oracle calls to the integrand and to the non-normalized density function. We are interested in analyzing the integration error of optimal algorithms (or the complexity of the problem) with emphasis on the variability of the weight function. For a corresponding large class of problem instances we show that the complexity grows linearly in the variability, and the simple Monte Carlo method provides an almost optimal algorithm. Under additional geometric restrictions (mainly log-concavity) for the density functions, we establish that a suitable adaptive local Metropolis algorithm is almost optimal and outperforms any non-adaptive algorithm.     

Adaptive order polynomial algorithm in a multiwavelet representation scheme

We have developed a new strategy to reduce the storage requirements of a multivariate function in a multiwavelet framework. We propose that alongside the commonly used adaptivity in the grid refinement one can also vary the order of the representation k as a function of the scale n. In particular the order is decreased with increasing refinement scale. The consequences of this choice, in particular with respect to the nesting of scaling spaces, are discussed and the error of the approximation introduced is analyzed. The application of this method to some examples of mono- and multivariate functions shows that our algorithm is able to yield a storage reduction up to almost 60%. In general, values between 30 and 40% can be expected for multivariate functions. Monovariate functions are less affected but are also much less critical in view of the so called “curse of dimensionality”.     

An Adaptive Uniaxial Perfectly Matched Layer Method for Time-Harmonic Scattering Problems

The uniaxial perfectly matched layer (PML) method uses rectangular domain to define the PML problem and thus provides greater flexibility and efficiency in deal- ing with problems involving anisotropic scatterers.In this paper an adaptive uniaxial PML technique for solving the time harmonic Helmholtz scattering problem is devel- oped.The PML parameters such as the thickness of the layer and the fictitious medium property are determined through sharp a posteriori error estimates.The adaptive finite element method based on a posteriori error estimate is proposed to solve the PML equa- tion which produces automatically a coarse mesh size away from the fixed domain and thus makes the total computational costs insensitive to the thickness of the PML absorb- ing layer.Numerical experiments are included to illustrate the competitive behavior of the proposed adaptive method.In particular,it is demonstrated that the PML layer can be chosen as close to one wave-length from the scatterer and still yields good accuracy and efficiency in approximating the far fields.     

Mechanically Interlocked [2]Rotaxane Aerogels with Tunable Morphologies and Mechanical Properties

Mechanical bonds have been utilized as promising motifs to construct mechanically interlocked aerogels (MIAs) with mechanical adaptivity and multifunctionality. However, fabricating such aerogels with not only precise chemical structures but also dynamic features remains challenging. Herein, we present MIAs carrying dense [2]rotaxane units, which bestow both the stability and flexibility of the aerogel network. Owing to the stable chemical structure of a [2]rotaxane, MIAs possessing a precise and full-scale mechanically interlocked network could be fabricated with the aid of diverse solvents. In addition, the dynamic nature of the [2]rotaxane resulted in morphologies and mechanical performances of the MIAs that can be dramatically modulated under chemical stimuli. We hope that the structure–property relationship in MIAs will facilitate the development of mechanically interlocked materials and provide novel opportunities toward constructing smart materials with multifunctionalities.     

Adaptive Catalysts for the Selective Hydrogenation of Bicyclic Heteroaromatics using Ruthenium Nanoparticles on a CO2-Responsive Support

Ruthenium nanoparticles (NPs) immobilized on an amine-functionalized polymer-grafted silica support act as adaptive catalysts for the hydrogenation of bicyclic heteroaromatics. Whereas full hydrogenation of benzofuran and quinoline derivatives is achieved under pure H₂, introducing CO₂ into the H₂ gas phase leads to an effective shutdown of the arene hydrogenation while preserving the activity for the hydrogenation of the heteroaromatic part. The selectivity switch originates from the generation of ammonium formate species on the surface of the materials by catalytic hydrogenation of CO₂. The CO₂ hydrogenation is fully reversible, resulting in a robust and rapid switch between the two states of the catalyst adapting its performance in response to the feed gas composition. A variety of benzofuran and quinoline derivatives were hydrogenated to fully or partially saturated products in high selectivity and yields simply by altering the composition of the feed gas from H₂ to H₂/CO₂. The adaptive catalytic system thus provides controlled access to valuable products using a single catalyst rather than two specific and distinct catalysts with static reactivity.     

Sampling schemes and recovery algorithms for functions of few coordinate variables

When a multivariate function does not depend on all of its variables, it can be approximated from fewer point evaluations than otherwise required. This has been previously quantified e.g. in the case where the target function is Lipschitz. This note examines the same problem under other assumptions on the target function. If it is linear or quadratic, then connections to compressive sensing are exploited in order to determine the number of point evaluations needed for recovering it exactly. If it is coordinatewise increasing, then connections to group testing are exploited in order to determine the number of point evaluations needed for recovering the set of active variables. A particular emphasis is put on explicit sets of evaluation points and on practical recovery methods. The results presented here also add a new contribution to the field of group testing.     

Adaptive rational spectral methods for the linear stability analysis of nonlinear fourth-order problems

This paper presents the application of adaptive rational spectral methods to the linear stability analysis of nonlinear fourth-order problems. Our model equation is a phase-field model of infiltration, but the proposed discretization can be directly extended to similar equations arising in thin film flows. The sharpness and structure of the wetting front preclude the use of the standard Chebyshev pseudo-spectral method, due to its slow convergence in problems where the solution has steep internal layers. We discuss the effectiveness and conditioning of the proposed discretization, and show that it allows the computation of accurate traveling waves and eigenvalues for small values of the initial water saturation/film precursor, several orders of magnitude smaller than the values considered previously in analogous stability analyses of thin film flows, using just a few hundred grid points.