董兆楠 研究员（法国国家信息与自动化研究所）

郑伟英 研究员

hp-A Posteriori Error Analysis of dG Methods for Maxwell and H(div)-conforming dG Methods for Stokes Problems via Novel Conforming Reconstruction Operators

11月3日（周二）16:00-17:00，南楼202

High-order discontinuous Galerkin methods provide flexible and accurate approximations 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 Poincaré inequality 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 Bogovskiĭ operator, 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.

References:

1.  Z. Dong and A. Ern.  H(curl)-reconstruction of piecewise polynomial fields with application to a posteriori nonconforming error analysis for Maxwell's equations. SIAM Journal on Numerical Analysis, 64(4), 1390–1413 (2026).

2.  Z. Dong, Z. Wang and L. Zhao.  Pressure-robust $hp$-a posteriori error estimates of H(div)-conforming discontinuous Galerkin methods for the Stokes equations. Submitted. Arxiv: 2607.04787

报告人简介：董兆楠，法国国家信息与自动化研究所（INRIA, Paris）研究员，主要研究方向包括continuous and discontinuous FEM、hp-version FEM、adaptive algorithms、multiscale methods、 polygonal discretization methods、hybrid high-order methods、solver design等。2016年10月在英国University of Leicester师从间断有限元方法专家Emmanuil Georgoulis教授和 Andrea Cangiani博士，获得应用数学博士，随后在英国University of Leicester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月至2020年9月在英国Cardiff University 数学系担任讲师。近年来在 SIAM Journal of Numerical Analysis、Mathematics of Computation、SIAM Journal on Scientific Computing等计算数学权威期刊上发表学术论文20余篇，并在Springer 出版社出版一部专著。