Maximum-norm stability of the finite element method for the Neumann problem in nonconvex polygons with locally refined mesh

The Galerkin finite element solution uh of the Possion equation −∆u = 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: ‖uh‖L∞(Ω) ≤ C`h‖u‖L∞(Ω), where `h = ln(2+1/h) for piecewise linear elements and `h = 1 for higher-order elements. As a result of the maximum-norm stability, the following best approximation result holds: ‖u− uh‖L∞(Ω) ≤ C`h‖u− Ihu‖L∞(Ω), where Ih 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− uh‖L∞(Ω) ≤ { C`hh k+2− 2 p ‖f‖Wk,p(Ω) if r ≥ k + 1, C`hh ‖f‖Hk(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.