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Journal of Computational Mathematics 2012年第2期30卷 101-123页

作者： M. hintermuer Michael hinze ronald h.w. hoppe Department of Mathematics Humboldt-University of Berlin Germany Department of Mathematics University of Hamburg Germany Department of Mathematics University of Augsburg Germany and Department of MathematicsUniversity of Houston USA

Adaptive finite element methods for optimization problems for second order linear el- liptic partial differential equations subject to pointwise constraints on the l2-norm of the gradient of the state are considered. In a weak duality setting, i.e. without assuming a constraint qualification such as the existence of a Slater point, residual based a posteriori error estimators are derived. To overcome the lack in constraint qualification on the continuous level, the weak Fenchel dual is utilized. Several numerical tests illustrate the performance of the proposed error *** subject classification： 65N30, 90C46, 65N50, 49K20, 49N15, 65K10.

关键词： Adaptive finite element method A posteriori errors Dualization Low regu-larity Pointwise gradient constraints State constraints weak solutions.

ERROR REDUCTION IN ADAPTIVE FINITE ELEMENT APPROXIMATIONS OF ELLIPTIC OBSTACLE PROBLEMS

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Journal of Computational Mathematics 2009年第2期27卷 148-169页

作者： Dietrich Braess Carsten Carstensen ronald h.w.hoppe Faculty of Mathematics Ruhr University BochumD-44780 BochumGermany Institute of Mathematics Humboldt University of BerlinD-10099 BerlinGermany Department of Mathematics University of HoustonHoustonTX 77204-3008USA Institute of Mathematics University at AugsburgD-86159 AugsburgGermany

we consider an adaptive finite element method （AFEM） for obstacle problems associated with linear second order elliptic boundary value problems and prove a reduction in the energy norm of the discretization error which leads to R-linear convergence. This result is sh.w. to hold up to a consistency error due to the extension of the discrete multipliers （point functionals） to h^-1 and a possible mismatch between the continuous and discrete coincidence and noncoincidence sets. The AFEM is based on a residual-type error estimator consisting of element and edge residuals. The a posteriori error analysis reveals that the significant difference to the unconstrained case lies in the fact that these residuals only have to be taken into account within the discrete noncoincidence set. The proof of the error reduction property uses the reliability and the discrete local efficiency of the estimator as well as a perturbed Galerkin orthogonality. Numerical results are given illustrating the performance of the AFEM.

关键词： Adaptive finite element methods Elliptic obstacle problems Convergence analysis

On Poincare-Friedrichs Type Inequalities for the Broken Sobolev Space w^(2,1)

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Numerical Mathematics(Theory,Methods and Applications) 2021年第1期14卷 31-46页

作者： R.h.w.hoppe Department of Mathematics University of HoustonUSA Department of Mathematics University of AugsburgGermany

we are concerned with the derivation of Poincare-Friedrichs type inequalities in the broken Sobolev space w^(2,1)(Ω;T h.w.th respect to a geometrically conforming,simplicial triagulation T h of a bounded Lipschitz domainΩin R d,d∈*** inequalities are of interest in the numerical analysis of nonconforming finite element discretizations such as C^(0) Discontinuous Galerkin(C^(0)DG)approximations of minimization problems in the Sobolev space w^(2,1)(Ω),or more generally,in the Banach space BV^(2)(Ω)of functions of bounded second order total *** an application,we consider a C^(0) DG approximation of a minimization problem in BV^(2)(Ω)which is useful for texture analysis and management in image restoration.

关键词： Poincare-Friedrichs inequalities broken Sobolev spaces C^(0)Discontinuous Galerkin approximation image processing

daptive hybridized Interior Penalty Discontinuous Galerkin Methods for h(curl)-Elliptic Problems

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Numerical Mathematics(Theory,Methods and Applications) 2011年第1期4卷 13-37页

作者： C.Carstensen R.h.w.hoppe N.Sharma T.warburton Department of Mathematics Humboldt Universität zu BerlinD-10099 BerlinGermany Department of Computer Science Engineering Yonsei UniversitySeoul 120-749Korea Department of Mathematics University of HoustonHouston TX 77204-3008USA Institute of Mathematics University of AugsburgD-86159 AugsburgGermany CAAM Rice UniversityHoustonTX 77005-1892USA.

we develop and analyze an adaptive hybridized Interior Penalty Discontinuous Galerkin(IPDG-h)method for h(curl)-elliptic boundary value problems in 2D or 3D arising from a semi-discretization of the eddy currents *** method can be derived from a mixed formulation of the given boundary value problem and involves a Lagrange multiplier that is an approximation of the tangential traces of the primal variable on the interfaces of the underlying triangulation of the computational *** is sh.w. that the IPDG-h technique can be equivalently formulated and thus implemented as a mortar *** mesh adaptation is based on a residual-type a posteriori error estimator consisting of element and face *** a unified framework for adaptive finite element methods,we prove the reliability of the estimator up to a consistency *** performance of the adaptive symmetric IPDG-h method is documented by numerical results for representative test examples in 2D.

关键词： Adaptive hybridized Interior Penalty Discontinuous Galerkin method a posteriori error analysis h(curl)-elliptic boundary value problems semi-discrete eddy currents equations

MODELING,SIMULATION,AND OPTIMIZATION OF SURFACE ACOUSTIC wAVE DRIVEN MICROFLUIDIC BIOChIPS

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Journal of Computational Mathematics 2010年第2期28卷 149-169页

作者： harbir Antil Roland Glowinski ronald h.w.hoppe Christopher Linsenmann Tsorng-whay Pan Achim wixforth Department of Mathematics University of Houston Institute of Mathematics University of Augsburg Institute of Physics University of Augsburg

we will be concerned with the mathematical modeling, numerical simulation, and shape optimization of micro fluidic biochips that are used for various biomedical applications. A particular feature is that the fluid flow in the fluidic network on top of the biochips is in- duced by surface acoustic waves generated by interdigital transducers. we are thus faced with a multiphysics problem that will be modeled by coupling the equations of piezoelectricity with the compressible Navier-Stokes equations. Moreover, the fluid flow exhibits a multiscale character that will be taken care of by a homogenization approach. we will discuss and analyze the mathematical models and deal with their numerical solution by space-time discretizations featuring appropriate finite element approximations with respect to hierarchies of simplicial triangulations of the underlying computational domains. Simulation results will be given for the propagation of the surface acoustic waves on top of the piezoelectric substrate and for the induced fluid flow in the microchannels of the fluidic network. The performance of the operational behavior of the biochips can be significantly improved by shape optimization. In particular, for such purposes we present a multilevel interior point method relying on a predictor-corrector strategy with an adaptive choice of the continuation steplength along the barrier path. As a specific example, we will consider the shape optimization of pressure driven capillary barriers between microchannels and reservoirs.

关键词： Microfluidic biochips Mathematical modeling Numerical simulation Shape optimization Multiphysics Multiscale problems.

OPTIMALITY OF LOCAL MULTILEVEL METhODS FOR ADAPTIVE NONCONFORMING P1 FINITE ELEMENT METhODS

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Journal of Computational Mathematics 2013年第1期31卷 22-46页

作者： Xuejun Xu huangxin Chen R.h.w. hoppe LSEC ICMSEC Academy of Mathematics and System Sciences Chinese Academy of Sciences Beijing 100190 China School of Mathematical Sciences Xiamen University Xiamen 361005 China LSEC ICMSEC Academy of Mathematics and System Sciences Chinese Academy of Sciences Beijing 100190 China Department of Mathematics University of Houston Houston TX 77204-3008 USA Institute for Mathematicd University of Augsburg D-86159 Augsburg Germany

In this paper, a local multilevel product algorithm and its additive version are con- sidered for linear systems arising from adaptive nonconforming P1 finite element approx- imations of second order elliptic boundary value problems. The abstract Schwarz theory is applied to analyze the multilevel methods with Jaeobi or Gauss-Seidel smoothers per- formed on local nodes on coarse meshes and global nodes on the finest mesh. It is sh.w. that the local multilevel methods are optimal, i.e., the convergence rate of the multilevel methods is independent of the mesh sizes and mesh levels. Numerical experiments are given to confirm the theoretical results.

关键词： Local multilevel methods Adaptive nonconforming P1 finite element methods,Convergence analysis Optimality.

CONVERGENCE OF ADAPTIVE EDGE ELEMENT METhODS FOR ThE 3D EDDY CURRENTS EQUATIONS

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Journal of Computational Mathematics 2009年第5期27卷 657-676页

作者： R.h.w.hoppe J.Schberl Institut fur Mathematik Universitat AugsburgD-86159AugsburgGermanyand Department of MathematicsUniversity of HoustonHoustonTX 77204-3008USA Dept.for Mathematics CCES(Center for Computational Engineering Science) RWTH AachenD-52074 AachenGermany

we consider an Adaptive Edge Finite Element Method （AEFEM） for the 3D eddy currents equations with variable coefficients using a residual-type a posteriori error estimator. Both the components of the estimator and certain oscillation terms, due to the occurrence of the variable coefficients, have to be controlled properly within the adaptive loop which is taken care of by appropriate bulk criteria. Convergence of the AEFEM in terms of reductions of the energy norm of the discretization error and of the oscillations is sh.w.. Numerical results are given to illustrate the performance of the AEFEM.

关键词： Adaptive edge elements 3D eddy currents equations Convergence analysis,Error and oscillation reduction Residual type a posteriori error estimates.

Approximation of Nearest Common Fixed Point of Nonexpansive Mappings in hilbert Spaces

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Acta Mathematica Sinica,English Series 2007年第10期23卷 1889-1896页

作者： Shi Sheng ZhANG h.w. JOSEPh LEE Chi Kin ChAN Department of Mathematics Yibin University Yibin 644007 P. R. China Department of Mathematics Sichuan University Chengdu 610064 P. R. China Department of Applied Mathematics The Hong Kong Polytechnic University Hung Horn Kowloon Hong Kong P. R. China

The purpose of this paper is to study the convergence problem of the iteration scheme xn＋l = λn＋1y ＋（1 - λn＋1）Tn＋1xn for a family of infinitely many nonexpansive mappings T1, T2,... in a hilbert space. It is proved that under suitable conditions this iteration scheme converges strongly to the nearest common fixed point of this family of nonexpansive mappings. The results presented in this paper extend and improve some recent results.

关键词： common fixed point family of nonexpansive mappings nearest point projection iterative scheme

Gas Quenching during Age hardening of the Aluminium Casting Alloy Al-7Si-0.3Mg

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材料热处理学报 2004年第5期25卷 498-501页

作者： A. Rose O. Kessler F. hoffmann h.w. Zoch Stiftung Institutfttr werkstofftechnik Bremen GermanyStiftung Institutfttr Werkstofftechnik Bremen GermanyStiftung Institutfttr Werkstofftechnik Bremen GermanyStiftung Institutfttr Werkstofftechnik Bremen Germany

For quenching of age hardenable aluminium alloys today predominantly aqueous quenching media are used, which can lead due to the Leidenfrost phenomenon to a non-uniform cooling of the parts and thus to thermal stresses. Particularly at thin-walled or complex shaped parts local plastic deformations can occur by the uneven thermal stresses. In relation to the conventional quenching procedures in aqueous media, gas quenching exhibits a number of technological, ecological and economical advantages. In comparison to liquid quenching media, gas does not change its phase during quenching. Moreover, the cleaning problem of the parts can be avoided. The quenching intensity can be adjusted by the variable parameters gas pressure and gas speed as well as the kind of gas and thus can be adapted to the requirements of the alloy. By the higher uniformity and the better reproducibility, gas quenching offers a high potential to reduce distortion. The goal of these investigations is to clarify, if the cooling rate during gas quenching is sufficient to obtain the specific required strength after age hardening of the alloy Al-7Si-0.3Mg. For this purpose different tests in high-pressure gas quenching facilities, gas nozzle fields and water quenching baths were performed.

关键词：气体流淬火铝合金 Al-7Si-0.3Mg 沉淀硬化机械性质

PREDICTION OF ThE MIXING ENThALPIES OF BINARY LIQUID ALLOYS BY MOLECULAR INTERACTION VOLUME MODEL

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Acta Metallurgica Sinica(English Letters) 2008年第5期21卷 336-340页

作者： h.w. Yang D.P. Tao Z.h. Zhou Faculty of Materials and Metallurgical Engineering Kunming University of Science and Technology Kunming 650093 China

The mixing enthalpies of 23 binary liquid alloys are calculated by molecular interaction volume model （MIVM）, which is a two-parameter model with the partial molar infinite dilute mixing enthalpies. The predicted va... 详细信息

关键词： Molecular interaction volume model Mixing enthalpy Liquid alloys Prediction