题目：Maximum-Norm Stability of the Finite Element Method for the Neumann Problem In Nonconvex Polygons with Locally Refined Mesh

报告人：李步扬 教授 （香港理工大学）

地点：致远楼101室

时间：2026年1月8日  15:00—15:45

摘要： The Galerkin finite element solutionu_hof the Possion equation -Δu=f under the Neumann boundary condition in a possibly nonconvex polygon Ω, with a graded mesh locally refined at the corners of the domain, is shown to satisfy the following maximum-norm stability:

‖u_h ‖_(L^∞ (Ω))≤Cl_h ‖u‖_(L^∞ (Ω)),

where l_h=ln⁡( 2+1/h) for piecewise linear elements and l_h=1 for higher-order elements. As a result of the maximum-norm stability, the following best approximation result holds:

‖u-u_h ‖_(L^∞ (Ω))≤Cl_h ‖u-I_h u‖_(L^∞ (Ω)),

where I_h denotes the Lagrange interpolation operator onto the finite element space. For a locally quasi-uniform triangulation sufficiently refined at the corners, the above best approximation property implies the following optimal-order error bound in the maximum norm:

‖u-u_h ‖_(L^∞ (Ω))≤{■(&Cl_h h^(k+2-2/p) ‖f‖_(W^(k,p) (Ω))&&if r≥k+1,@&Cl_h h^(k+1) ‖f‖_(H^k (Ω))&&if r=k,)┤

where r≥1 is the degree of finite elements, k is any nonnegative integer no larger than r, and p∈[2,∞) can be arbitrarily large.

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