各向异性网格下二阶椭圆问题的一个混合元形式

提出了二阶椭圆问题的一个混合变分形式,同时证明了Rariart-Thomas元的各向异性插值性质,并给出了单元的对二阶问题的最优误差估计。     

The Fourier-finite-element method with Nitsche mortaring

^** Email: bernd.heinrich{at}mathematik.tu-chemnitz.de The paper deals with a combination of the Fourier-finite-elementmethod with the Nitsche-finite-element method (as a mortar method).The approach is applied to the Dirichlet problem for the Poissonequation in 3D axisymmetric domains with non-axisymmetric data.The approximating Fourier method yields a splitting of the 3Dproblem into 2D problems on the meridian plane of the givendomain. For solving these 2D problems, the Nitsche-finite-elementmethod with non-matching meshes is applied. Some important propertiesof the approximation scheme are derived and the rate of convergencein an H¹-like norm as well as in the L₂-norm is estimated fora regular solution. Finally, some numerical results are presented.     

Exponential splitting time integration for pseudospectral methods on moving meshes

Moving meshes are successfully used in many fields. Here we investigate how a recently proposed approach to combine the Strang splitting method for time integration with pseudospectral spatial discretization by orthogonal polynomials can be extended to include moving meshes. A double representation of a function (by coefficients of polynomial expansion and by values at the mesh nodes associated with a suitable quadrature formula) is an essential part of the numerical integration. Before numerical implementation the original PDE is transformed into a suitable form. The approach is illustrated on the linear heat transfer equation.     

油气资源数值模拟的变网格交替方向特征有限元格式和分析

石油科学在有机地球化学、石油的生成、运移、聚集研究取得重大进展,在评价油气资源时,对盆地发育史尤其对流体流动规律和受热变化历史的计算是非常重要的.其数学模型是三维空间非线性偶合偏微分方程组的初边值问题.从实际出发,考虑了流体的压缩性和三维问题大规模科学与工程计算的特征, 提出了一类变网格交替方向特征有限元格式,应用变分形式、算子分裂、广义$L^2$投影、能量方法、负模估计、微分方程先验估计的理论和技巧,得到最佳阶$L^2$误差估计. 此方法已成功应用到油气资源评估数值模拟的生产实践中,成功解决了这一重要问题.     

Convergence and superconvergence analysis of a new quadratic Hermite-type triangular element on anisotropic meshes

A new quadratic Hermite-type triangular finite element is conceived to solve a class of two-dimensional second-order elliptic boundary value problems. Its error estimates on anisotropic meshes are developed. Furthermore, we verify that some conditions set to the meshes contribute to the proof of its superconvergence properties, which can improve the approximation results. Numerical examples are given to confirm our theoretical analysis.     

A second order difference scheme with nonuniform rectangular meshes for nonlinear parabolic system

In this paper, a difference scheme with nonuniform meshes is proposed for the initial-boundary problem of the nonlinear parabolic system. It is proved that the difference scheme is second order convergent in both space and time.     

Multimodal Biomedical Implant with Plasmonic and Simulated Body Temperature Responses

This work presents a novel nanoparticle-based thermosensor implant able to reveal the precise temperature variations along the polymer filaments, as it contracts and expands due to changes in the macroscale local temperature. The multimodal device is able to trace the position and the temperature of a polypropylene mesh, employed in abdominal hernia repair, by combining plasmon resonance and Raman spectroscopy with hydrogel responsive system. The novelty relies on the attachment of the biocompatible nanoparticles, based on gold stabilized by a chitosan-shell, already charged with the Raman reporter (RaR) molecules, to the robust prosthesis, without the need of chemical linkers. The SERS enhanced effect observed is potentiated by the presence of a quite thick layer of the copolymer (poly(N-isopropylacrylamide)-co-poly(acrylamide)) hydrogel. At temperatures above the LCST of PNIPAAm-co-PAAm, the water molecules are expulsed and the hydrogel layer contracts, leaving the RaR molecules more accessible to the Raman source. In vitro studies with fibroblast cells reveal that the functionalized surgical mesh is biocompatible and no toxic substances are leached in the medium. The mesh sensor opens new frontiers to semi-invasive diagnosis and infection prevention in hernia repair by using SERS spectroscopy. It also offers new possibilities to the functionalization of other healthcare products.