H(curl)-reconstruction of piecewise polynomial fields
with application to hp-a posteriori error analysis
for dG methods for Maxwell's equations
报告专家：董兆楠 研究员 （法国国家信息与自动化研究所（INRIA，Paris））
报告时间：3月16日（周一)10:00-11:00
报告地点：国家天元数学西南中心516
报告摘要：
We devise and analyse a novel $\boldsymbol{H}(\textbf{curl)$-reconstruction operator for piecewise polynomial fields on shape-regular simplicial meshes. The (non-polynomial) reconstruction is devised over the mesh vertex patches using the partition of unity induced by hat basis functions in combination with local Helmholtz decompositions. Our main focus is on homogeneous tangential boundary conditions. We prove that the difference between the reconstructed $\boldsymbol{H}_0(\textbf{curl})$-field and the original, piecewise polynomial field, measured in the broken curl norm and in the $\boldsymbol{L}^2$-norm, can be bounded in terms of suitable jump norms of the original field. The bounds are always $h$-optimal, and $p$-suboptimal by $1/2$-order for the broken curl norm and by $3/2$-order for the $\boldsymbol{L}^2$-norm. An auxiliary result of independent interest is a novel broken-curl, divergence-preserving Poincar\'{e} inequality on vertex patches. Moreover, the $\boldsymbol{L}^2$-norm estimate can be improved to $\frac12$-order suboptimality under a (reasonable) assumption on the uniform elliptic regularity pickup for a Poisson problem with Neumann conditions over the vertex patches. We also discuss extensions of the $\boldsymbol{H}_0(\textbf{curl})$-reconstruction operator to the prescription of mixed boundary conditions, to agglomerated polytopal meshes, and to convex domains. Finally, we showcase an important application of the $\boldsymbol{H}(\textbf{curl})$-reconstruction operator to the $hp$-a posteriori nonconforming error analysis of Maxwell's equations. We focus on the (symmetric) interior penalty discontinuous Galerkin (dG) approximation of some simplified forms of Maxwell's equations.
专家简介：
董兆楠, 法国国家信息与自动化研究所（INRIA，Paris）研究员(Tenured Researcher), 主要研究方向包括：continuous and discontinuous FEM, hp-version FEM, adaptive algorithms, a posteriori error analysis, multiscale methods, polygonal discretization methods, hybrid high-order methods, solver design. 2016年10月在英国University of Leicester师从于间断有限元有限元方法（dG）的专家Emmanuil Georgoulis教授和 Andrea Cangiani教授，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-2020年9月在英国Cardiff University 数学系做讲师。2020起在法国INRIA研究所开始做研究员。近年来在 SIAM Journal of Numerical Analysis, Mathematics of Computation, SIAM Journal on Scientific Computing等杂志上发表30余篇文章，并在Springer 出版社出版一部专著。
邀请人：郭汝驰
