Regularity and numerical approximation of fractional elliptic differential equations on compact metric graphs
David Bolin, Mihály Kovács, Vivek Kumar, Alexandre B. Simas
Abstract
The fractional differential equation <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper L Superscript beta Baseline u equals f"> <mml:semantics> <mml:mrow> <mml:msup> <mml:mi>L</mml:mi> <mml:mi> β </mml:mi> </mml:msup> <mml:mi>u</mml:mi> <mml:mo>=</mml:mo> <mml:mi>f</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">L^\beta u = f</mml:annotation> </mml:semantics> </mml:math> </inline-formula> posed on a compact metric graph is considered, where <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="beta greater-than 0"> <mml:semantics> <mml:mrow> <mml:mi> β </mml:mi> <mml:mo>></mml:mo> <mml:mn>0</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">\beta >0</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper L equals kappa squared minus nabla left-parenthesis a nabla right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>L</mml:mi> <mml:mo>=</mml:mo> <mml:msup> <mml:mi> κ </mml:mi> <mml:mn>2</mml:mn> </mml:msup> <mml:mo> − </mml:mo> <mml:mi mathvariant="normal"> ∇ </mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>a</mml:mi> <mml:mi mathvariant="normal"> ∇ </mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">L = \kappa ^2 - \nabla (a\nabla )</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is a second-order elliptic operator equipped with certain vertex conditions and sufficiently smooth and positive coefficients <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="kappa comma a"> <mml:semantics> <mml:mrow> <mml:mi> κ </mml:mi> <mml:mo>,</mml:mo> <mml:mi>a</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\kappa ,a</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . We demonstrate the existence of a unique solution for a general class of vertex conditions and derive the regularity of the solution in the specific case of Kirchhoff vertex conditions. These results are extended to the stochastic setting when <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="f"> <mml:semantics> <mml:mi>f</mml:mi> <mml:annotation encoding="application/x-tex">f</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is replaced by Gaussian white noise. For the deterministic and stochastic settings under generalized Kirchhoff vertex conditions, we propose a numerical solution based on a finite element approximation combined with a rational approximation of the fractional power <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper L Superscript negative beta"> <mml:semantics> <mml:msup> <mml:mi>L</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo> − </mml:mo> <mml:mi> β </mml:mi> </mml:mrow> </mml:msup> <mml:annotation encoding="application/x-tex">L^{-\beta }</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . For the resulting approximation, the strong error is analyzed in the deterministic case, and the strong mean squared error as well as the <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper L 2 left-parenthesis normal upper Gamma times normal upper Gamma right-parenthesis"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi>L</mml:mi> <mml:mn>2</mml:mn> </mml:msub> <mml:mo stretchy="false">(</mml:mo> <mml:mi mathvariant="normal"> Γ </mml:mi> <mml:mo> × </mml:mo> <mml:mi mathvariant="normal"> Γ </mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">L_2(\Gamma \times \Gamma )</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -error of the covariance function of the solution are analyzed in the stochastic setting. Explicit rates of convergences are derived for all cases. Numerical experiments for <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper L equals kappa squared minus normal upper Delta comma kappa greater-than 0"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>L</mml:mi> <mml:mo>=</mml:mo> <mml:msup> <mml:mi> κ </mml:mi> <mml:mn>2</mml:mn> </mml:msup> <mml:mo> − </mml:mo> <mml:mi mathvariant="normal"> Δ </mml:mi> <mml:mo>,</mml:mo> <mml:mi> κ </mml:mi> <mml:mo>></mml:mo> <mml:mn>0</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">{L = \kappa ^2 - \Delta , \kappa >0}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> are performed to illustrate the results.