报告人：董兆楠 研究员 法国国家信息与自动化研究所
报告题目：hp-A Posteriori Error Analysis of dG Methods for Maxwell and H(div)-conforming dG Methods for Stokes Problems via Novel Conforming Reconstruction Operators
摘 要：High-order discontinuous Galerkin methods provide flexible and accurate
approxi-mations of vector-valued PDEs, but their nonconformity poses substantial challenges for a posteriori error analysis. In this talk, we present conforming reconstruction techniques and their applications to (hp)-explicit error estimation for Maxwell’s equations and the Stokes problem. We first introduce a novel (\boldsymbol{H}(\mathrm{curl}))-reconstruction operator for piecewise polynomial vector fields on shape-regular simplicial meshes. The reconstruction is built locally over vertex patches by combining the partition of unity induced by standard hat functions with local Helmholtz decompositions. Its distance from the original field, measured in both the broken curl norm and the (\boldsymbol{L}^2)-norm, is controlled by suitable tangential-jump indicators. The resulting estimates are optimal in the mesh size and nearly optimal in the polynomial degree. A key analytical ingredient is a new divergence-preserving broken Poincaréinequality on vertex patches. This reconstruction is then applied to the (hp)-a posteriori analysis of symmetric interior penalty DG approximations of Maxwell’s equations. We next consider (\boldsymbol{H}(\mathrm{div}))-conforming DG methods for the Stokes equations. By combining a conforming-nonconforming error decomposition, local Helmholtz decompositions, a generalized Bogovskiĭoperator, and specially designed (H^2)-bubble functions, we derive residual-based estimators providing guaranteed upper bounds and local lower bounds for the velocity error. Crucially, these estimates are independent of the pressure and robust with respect to the viscosity. Together, these results illustrate how local conforming reconstructions and differential-complex structures can provide a systematic foundation for reliable and robust adaptive methods for vector-valued PDEs.
报告人简介：董兆楠研究员(Tenured Researcher)，法国国家信息与自动化研究所(INRIA, Paris)，主要研究方向包括：continuous and discontinuous FEM, hp-version FEM, adaptive algorithms, multiscale methods, hybrid high-order methods, a posteriori error estimates and solver design. 2016年10月于英国莱斯特大学(University of Leicester)获应用数学博士学位，师从于间断有限元(dG)专家Emmanuil Georgoulis教授和Andrea Cangiani教授。随后在英国莱斯特大学从事博士后研究。2019年在希腊国立数学研究院(IACM-FORTH)做访问学者。2019年11月-2020年9月在英国卡迪夫大学(Cardiff University)数学系任讲师。2020年起在法国INRIA研究所任研究员。近年来在SIAM Journal of Numerical Analysis, Mathematics of Computation, Numerische Mathematik, SIAM Journal on Scientific Computing等国际顶刊发表多篇论文。
报告时间：2026年10月23日，15:00--16:30
报告地点：玉泉校区创新北楼512会议室
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内蒙古大学数学科学学院
2026年10月16日
蒙ICP16002391号-1