AFEM for the Laplace-Beltrami operator on graphs: Design and conditional contraction property

Abstract

We present an adaptive finite element method (AFEM) of any polynomial degree for the Laplace-Beltrami operator on <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper C Superscript 1"> <mml:semantics> <mml:msup> <mml:mi>C</mml:mi> <mml:mn>1</mml:mn> </mml:msup> <mml:annotation encoding="application/x-tex">C^1</mml:annotation> </mml:semantics> </mml:math> </inline-formula> graphs <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper Gamma"> <mml:semantics> <mml:mi mathvariant="normal"> Γ </mml:mi> <mml:annotation encoding="application/x-tex">\Gamma</mml:annotation> </mml:semantics> </mml:math> </inline-formula> in <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-struck upper R Superscript d Baseline left-parenthesis d greater-than-or-equal-to 2 right-parenthesis"> <mml:semantics> <mml:mrow> <mml:msup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">R</mml:mi> </mml:mrow> <mml:mi>d</mml:mi> </mml:msup> <mml:mtext> </mml:mtext> <mml:mo stretchy="false">(</mml:mo> <mml:mi>d</mml:mi> <mml:mo> ≥ </mml:mo> <mml:mn>2</mml:mn> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathbb {R}^d ~(d\ge 2)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . We first derive residual-type a posteriori error estimates that account for the interaction of both the energy error in <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper H Superscript 1 Baseline left-parenthesis normal upper Gamma right-parenthesis"> <mml:semantics> <mml:mrow> <mml:msup> <mml:mi>H</mml:mi> <mml:mn>1</mml:mn> </mml:msup> <mml:mo stretchy="false">(</mml:mo> <mml:mi mathvariant="normal"> Γ </mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">H^1(\Gamma )</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and the surface error in <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper W Subscript normal infinity Superscript 1 Baseline left-parenthesis normal upper Gamma right-parenthesis"> <mml:semantics> <mml:mrow> <mml:msubsup> <mml:mi>W</mml:mi> <mml:mi mathvariant="normal"> ∞ </mml:mi> <mml:mn>1</mml:mn> </mml:msubsup> <mml:mo stretchy="false">(</mml:mo> <mml:mi mathvariant="normal"> Γ </mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">W^1_\infty (\Gamma )</mml:annotation> </mml:semantics> </mml:math> </inline-formula> due to approximation of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper Gamma"> <mml:semantics> <mml:mi mathvariant="normal"> Γ </mml:mi> <mml:annotation encoding="application/x-tex">\Gamma</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . We devise a marking strategy to reduce the total error estimator, namely a suitably scaled sum of the energy, geometric, and inconsistency error estimators. We prove a conditional contraction property for the sum of the energy error and the total estimator; the conditional statement encodes resolution of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper Gamma"> <mml:semantics> <mml:mi mathvariant="normal"> Γ </mml:mi> <mml:annotation encoding="application/x-tex">\Gamma</mml:annotation> </mml:semantics> </mml:math> </inline-formula> in <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper W Subscript normal infinity Superscript 1"> <mml:semantics> <mml:msubsup> <mml:mi>W</mml:mi> <mml:mi mathvariant="normal"> ∞ </mml:mi> <mml:mn>1</mml:mn> </mml:msubsup> <mml:annotation encoding="application/x-tex">W^1_\infty</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . We conclude with one numerical experiment that illustrates the theory.