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 Poisson 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: (Formula Presented) where l _h = ln(2+1/h) for piecewise linear elements and l _h = 1 for higherorder elements. As a result of the maximum-norm stability, the following best approximation result holds: (Formula Presented) 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 optimalorder error bound in the maximum norm: (Formula Presented) 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.