报告题目：C1-conforming Petrov-Galerkin Methods for 2nd-order Eproblems and Superconvergence
报 告 人：张智民 教授，北京计算科学研究中心
报告时间：2021年03月29日（星期一） 14:30-15:30 (北京时间, +8 GMT)
报告平台：腾讯会议 ID: 287 899 062
校内联系人：王翔 wxjldx@jlu.edu.cn
报告摘要：For 2nd-order elliptic problem, we propose a C1 Petro-Galerkin method, in which kth-order $C^1$-conforming finite elements are adopted for the trial space, and k-2th order discontinuous ($C^{-1}$ or $L^2$) piecewise polynomials are used as the test space. This is in contrast to the classical $C^1$-conforming finite element methods when both trial and test spaces use $C^1$ continuous piecewise polynomials. There is another alternative, using $C^0$ continuous piecewise polynomials as the test space. However, both theoretical analysis and numerical test indicate that $C^1$-$L^2$ pair is superior to the $C^1$-$C^0$ pair in the Petrov-Galerkin method.
The advantage of the $C^1$-$L^2$ Petrov-Galekin method is that it approximates derivatives (gradient) much more accurately than its counterpart existing methods. We prove that at the element nodal points, numerical approximation for both function and its gradient converge at rate 2k-2. We also identify superconvergence points/lines inside elements for function, the first-order and second-order derivatives. Numerical test results demonstrate that our theoretical error bounds are sharp.
报告人简介：张智民，教授，北京计算科学研究中心应用与计算数学研究部主任， Charles H. Gershenson 杰出学者，世界华人数学家大会45分钟报告人（2010，2019），现任和曾任10余个国内外数学杂志编委，包括Mathematics of Computation、Journal of Scientific Computing、Numerical methods for Partial Differential Equations, Journal of Computational Mathematics、CSIAM Transaction on Applied Mathematics、《数学文化》等。发表SCI论文180余篇，论文google 引用4600余次。 张智民教授长期从事计算方法，尤其是有限元方法的研究，在超收敛、后验误差估计、自适应算法和PDE特征值计算等领域的开拓性研究取得了多项创新成果。其所提出的多项式保持重构（Polynomial Preserving Recovery，PPR）方法被广泛的研究和应用，2008年被大型商业软件COMSOL Multiphysics 采用。
