Error estimates with polynomial growth <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.svg"><mml:mi mathvariant="script">O</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:msup><mml:mrow><mml:mi>ε</mml:mi></mml:mrow><mml:mrow><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msup><mml:mo stretchy="false">)</mml:mo></mml:math> for the HHO method on polygonal meshes of the Allen-Cahn model
Naresh Kumar, Ajeet Singh, Ram Jiwari, Jinyun Yuan
Abstract
A novel approach is presented to tackle the Allen-Cahn equation arising from phase separation in alloys, by utilizing the hybrid high-order (HHO) method on polygonal meshes. The primary challenge in this equation lies in employing a straightforward Gronwall inequality-type argument for error estimation with exponential growth factor e ( C T / ε 2 ) as ε approaches zero. The application of the discrete Lyapunov functional and the discrete HHO spectrum estimate of the linearized Allen-Cahn operator λ A C H H O are used to overcome this exponential growth factor and achieve polynomial growth of order O ( ε − 1 ) for error bounds in error estimations. Rigorous convergence analyses are established for the fully implicit schemes, which are energy stable. However, due to the implicit processing of the nonlinear term , the computational cost significantly increases. To enhance computational efficiency, a static condensation process is hired by using the HHO method, resulting in optimal convergence rates in L 2 norm. Finally, various numerical experiments on diverse meshes are conducted to validate our theoretical findings.