Maximum-norm stability of the finite element method for the Neumann problem in nonconvex polygons with locally refined mesh
Authors: Li Buyang 
Published: January 2022

The Galerkin finite element solutionuhu_hof the Poisson equation−Δu=f-\Delta u=funder the Neumann boundary condition in a possibly nonconvex polygonΩ\varOmega, with a graded mesh locally refined at the corners of the domain, is shown to satisfy the following maximum-norm stability:<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartLayout 1st Row double-vertical-bar u Subscript h Baseline double-vertical-bar Subscript upper L Sub Superscript normal infinity Subscript left-parenthesis upper Omega right-parenthesis Baseline less-than-or-equal-to upper C script l Subscript h Baseline double-vertical-bar u double-vertical-bar Subscript upper L Sub Superscript normal infinity Subscript left-parenthesis upper Omega right-parenthesis Baseline comma EndLayout">‖uh‖L∞(Ω)≤Cℓh‖u‖L∞(Ω),\begin{align*} \|u_h\|_{L^{\infty }(\varOmega )} \le C\ell _h\|u\|_{L^{\infty }(\varOmega )} , \end{align*}whereℓh=ln⁡(2+1/h)\ell _h = \ln (2+1/h)for piecewise linear elements andℓh=1\ell _h=1for higher-order elements. As a result of the maximum-norm stability, the following best approximation result holds:<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartLayout 1st Row double-vertical-bar u minus u Subscript h Baseline double-vertical-bar Subscript upper L Sub Superscript normal infinity Subscript left-parenthesis upper Omega right-parenthesis Baseline less-than-or-equal-to upper C script l Subscript h Baseline double-vertical-bar u minus upper I Subscript h Baseline u double-vertical-bar Subscript upper L Sub Superscript normal infinity Subscript left-parenthesis upper Omega right-parenthesis Baseline comma EndLayout">‖u−uh‖L∞(Ω)≤Cℓh‖u−Ihu‖L∞(Ω),\begin{align*} \|u-u_h\|_{L^{\infty }(\varOmega )} \le C\ell _h\|u-I_hu\|_{L^{\infty }(\varOmega )} , \end{align*}whereIhI_hdenotes 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:<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartLayout 1st Row double-vertical-bar u minus u Subscript h Baseline double-vertical-bar Subscript upper L Sub Superscript normal infinity Subscript left-parenthesis upper Omega right-parenthesis Baseline less-than-or-equal-to StartLayout Enlarged left-brace 1st Row 1st Column upper C script l Subscript h Baseline h Superscript k plus 2 minus StartFraction 2 Over p EndFraction Baseline double-vertical-bar f double-vertical-bar Subscript upper W Sub Superscript k comma p Subscript left-parenthesis upper Omega right-parenthesis Baseline 2nd Column a m p semicolon if r greater-than-or-equal-to k plus 1 comma 2nd Row 1st Column upper C script l Subscript h Baseline h Superscript k plus 1 Baseline double-vertical-bar f double-vertical-bar Subscript upper H Sub Superscript k Subscript left-parenthesis upper Omega right-parenthesis Baseline 2nd Column a m p semicolon if r equals k comma EndLayout EndLayout">‖u−uh‖L∞(Ω)≤{Cℓhhk+2−2p‖f‖Wk,p(Ω)amp;if r≥k+1,Cℓhhk+1‖f‖Hk(Ω)amp;if r=k,\begin{align*} \|u-u_h\|_{L^\infty (\varOmega )} \le \begin {cases} C\ell _h h^{k+2-\frac {2}{p}} \|f\|_{W^{k,p}(\varOmega )} &\text {if $r\ge k+1$}, \\ C\ell _h h^{k+1} \|f\|_{H^{k}(\varOmega )} &\text {if $r=k$}, \end{cases} \end{align*}wherer≥1r\ge 1is the degree of finite elements,kkis any nonnegative integer no larger thanrr, andp∈[2,∞)p\in [2,\infty )can be arbitrarily large.