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Event(s) on August 2011
- Thursday, 11th August, 2011
Title: Local Oscillations in Finite Difference Solutions of Hyperbolic Conservation Laws Speaker: Prof. Huazhong TANG, School of Mathematical Sciences, Peking University, China Time/Place: 14:30 - 15:30
FSC1217, Fong Shu Chuen Library, HSH Campus, Hong Kong Baptist UniversityAbstract: It was generally expected that monotone schemes are oscillation-free for hyperbolic conservation laws. However, recently local oscillations were observed and usually understood to be caused by relative phase errors. In order to further explain this, we first investigate the discretization of initial data that trigger the chequerboard mode, the highest frequency mode. Then we proceed to use the discrete Fourier analysis and the modified equation analysis to distinguish the dissipative and dispersive effects of numerical schemes for low frequency and high frequency modes, respectively. It is shown that the relative phase error is of order ${mathcal O}(1)$ for the high frequency modes $u_j^n=lambda^n_k e^{ixi j}$, $xiapprox pi$, but of order ${mathcal O}(xi^2)$ for low frequency modes ($xi approx 0$). In order to avoid numerical oscillations, the relative phase errors should be offset by numerical dissipation of at least the same order. Numerical damping, i.e. the zero order term in the corresponding modified equation, is important to dissipate the oscillations caused by the relative phase errors of high frequency modes. This is in contrast to the role of numerical viscosity, the second order term, which is the lowest order term usually present to suppress the relative phase errors of low frequency modes. It is a joint work with Jiequan Li (School of Mathematics, Beijing Normal University, Beijing 100875, P.R. China) and Lumei Zhang (School of Mathematics, Capital Normal University, Beijing 100037, P.R. China), and Gerald Warnecke(Institut f"ur Analysis und Numerik, Otto-von-Guericke-Universit"at, PSF 4120, 39016 Magdeburg, F.R. Germany). - Tuesday, 16th August, 2011
Title: Coupled Atomistic and Continuum Modeling of Brittle Cracks Speaker: Prof. Zhijian YANG, School of Mathematics and Statistics, Wuhan University, China Time/Place: 11:30 - 12:30
FSC1217, Fong Shu Chuen Library. HSH Campus, Hong Kong Baptist UniversityAbstract: I will take brittle cracks as examples to address the issues in coupled atomistic and continuum modeling, with an emphasis on differences between interface conditions for static and dynamic problems. For Static problems, the issue can be resolved by imposing consistency conditions. The necessary and sufficient condition for uniform first-order accuracy and, consequently, the elimination of the “ghost force” is formulated in terms of the reconstruction schemes. Examples of reconstruction schemes that satisfy this condition are presented. Transition between atom-based and element-based summation rules will be studied. While for dynamic cases, one has to deal with wave reflections as well. I will present a multiscale model for numerical simulations of dynamics of crystalline solids. The method combines the continuum nonlinear elasto-dynamics model, which models the stress waves and physical loading conditions, and molecular dynamics model, which provides the nonlinear constitutive relation and resolves the atomic structures near local defects. The coupling of the two models is achieved based on a general framework for Multiscale modeling – the heterogeneous multiscale method (HMM). I will derive an explicit coupling condition at the atomistic/continuum interface. Application to the dynamics of brittle cracks under various loading conditions is presented as test examples. - Tuesday, 30th August, 2011
Title: Adaptive Finite Element Methods for H(curl) and H(div) Problems Speaker: Dr. Long Chen, Department of Mathematics, University of California at Irvine, USA Time/Place: 11:00 - 12:00
FSC1217, Fong Shu Chuen Library, HSH Campus, Hong Kong Baptist UniversityAbstract: We design adaptive finite element methods (AFEMs) for variational problems posed in the Hilbert spaces H(div) and H(curl) in two and three dimensions. The main difficulty is the large null space of curl or div operators and we solve it by using discrete regular decompositions and a novel stable and local projection operator. As a result, we obtain convergence and optimal complexity of our adaptive algorithms.