Maximum-norm stability of the finite element method for the Neumann problem in nonconvex polygons with locally refined mesh
Buyang Li
Abstract
The Galerkin finite element solution <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="u Subscript h"> <mml:semantics> <mml:msub> <mml:mi>u</mml:mi> <mml:mi>h</mml:mi> </mml:msub> <mml:annotation encoding="application/x-tex">u_h</mml:annotation> </mml:semantics> </mml:math> </inline-formula> of the Poisson equation <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="minus normal upper Delta u equals f"> <mml:semantics> <mml:mrow> <mml:mo> − </mml:mo> <mml:mi mathvariant="normal"> Δ </mml:mi> <mml:mi>u</mml:mi> <mml:mo>=</mml:mo> <mml:mi>f</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">-\Delta u=f</mml:annotation> </mml:semantics> </mml:math> </inline-formula> under the Neumann boundary condition in a possibly nonconvex polygon <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper Omega"> <mml:semantics> <mml:mi> Ω </mml:mi> <mml:annotation encoding="application/x-tex">\varOmega</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , with a graded mesh locally refined at the corners of the domain, is shown to satisfy the following maximum-norm stability: <disp-formula content-type="math/mathml"> <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"> <mml:semantics> <mml:mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" side="left" displaystyle="true"> <mml:mtr> <mml:mtd> <mml:mo fence="false" stretchy="false"> ‖ </mml:mo> <mml:msub> <mml:mi>u</mml:mi> <mml:mi>h</mml:mi> </mml:msub> <mml:msub> <mml:mo fence="false" stretchy="false"> ‖ </mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msup> <mml:mi>L</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="normal"> ∞ </mml:mi> </mml:mrow> </mml:msup> <mml:mo stretchy="false">(</mml:mo> <mml:mi> Ω </mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> </mml:msub> <mml:mo> ≤ </mml:mo> <mml:mi>C</mml:mi> <mml:msub> <mml:mi> ℓ </mml:mi> <mml:mi>h</mml:mi> </mml:msub> <mml:mo fence="false" stretchy="false"> ‖ </mml:mo> <mml:mi>u</mml:mi> <mml:msub> <mml:mo fence="false" stretchy="false"> ‖ </mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msup> <mml:mi>L</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="normal"> ∞ </mml:mi> </mml:mrow> </mml:msup> <mml:mo stretchy="false">(</mml:mo> <mml:mi> Ω </mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> </mml:msub> <mml:mo>,</mml:mo> </mml:mtd> </mml:mtr> </mml:mtable> <mml:annotation encoding="application/x-tex">\begin{align*} \|u_h\|_{L^{\infty }(\varOmega )} \le C\ell _h\|u\|_{L^{\infty }(\varOmega )} , \end{align*}</mml:annotation> </mml:semantics> </mml:math> </disp-formula> where <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script l Subscript h Baseline equals ln left-parenthesis 2 plus 1 slash h right-parenthesis"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi> ℓ </mml:mi> <mml:mi>h</mml:mi> </mml:msub> <mml:mo>=</mml:mo> <mml:mi>ln</mml:mi> <mml:mo> </mml:mo> <mml:mo stretchy="false">(</mml:mo> <mml:mn>2</mml:mn> <mml:mo>+</mml:mo> <mml:mn>1</mml:mn> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo>/</mml:mo> </mml:mrow> <mml:mi>h</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\ell _h = \ln (2+1/h)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> for piecewise linear elements and <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script l Subscript h Baseline equals 1"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi> ℓ </mml:mi> <mml:mi>h</mml:mi> </mml:msub> <mml:mo>=</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">\ell _h=1</mml:annotation> </mml:semantics> </mml:math> </inline-formula> for higher-order elements. As a result of the maximum-norm stability, the following best approximation result holds: