報告題目：C1-conforming Petrov-Galerkin Methods for 2nd-order Eproblems and Superconvergence
報 告 人：張智民 教授，北京計算科學研究中心
報告時間：2021年03月29日（星期一） 14:30-15:30 (北京時間, +8 GMT)
報告平台：騰訊會議 ID: 287 899 062
校内聯系人：王翔 wxjldx@jlu.edu.cn
報告摘要：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.
報告人簡介：張智民，教授，北京計算科學研究中心應用與計算數學研究部主任， Charles H. Gershenson 傑出學者，世界華人數學家大會45分鐘報告人（2010，2019），現任和曾任10餘個國内外數學雜志編委，包括Mathematics of Computation、Journal of Scientific Computing、Numerical methods for Partial Differential Equations, Journal of Computational Mathematics、CSIAM Transaction on Applied Mathematics、《數學文化》等。發表SCI論文180餘篇，論文google 引用4600餘次。 張智民教授長期從事計算方法，尤其是有限元方法的研究，在超收斂、後驗誤差估計、自适應算法和PDE特征值計算等領域的開拓性研究取得了多項創新成果。其所提出的多項式保持重構（Polynomial Preserving Recovery，PPR）方法被廣泛的研究和應用，2008年被大型商業軟件COMSOL Multiphysics 采用。
