Recovery Type A Posteriori Error Estimation Of Interior Penalty Discontinuous Galerkin Method For Allen-Cahn Equation

Allen-Cahn equation has emerged as a fundamental equation in the phase field methodology or the diffuse interface methodology for moving interface and free boundary problems arising from various applications such as fluid dynamics,materials science,image processing and biology.In this thesis,we consider the recovery baesd a posteriori error estimation of the modified Crank-Nicolson interior penalty discontinuous Galerkin method for Allen-Cahn equation.We mainly focus on a posteriori error estimation of the elliptic part,so we first analyze the posteriori error estimation of the second order nonlinear elliptic equation corresponding to the Allen-Cahn equation.Numerical experiments show that the proposed recovery type a posteriori error estimator can guide adaptive mesh refinement,and the error estimates are asymptotically accurate.We futher derive a posteriori error estimation of the space semidiscrete and fully discrete schemes for Allen-Cahn equation,in which the derivation of error is based on the elliptic reconstruction technique,and the non-conforming part of the error is estimated by an appropriate computable quantity.Under the condition that the numerical solution is bounded,the reliability and upper bound of the a posterior error estimator of interior penalty discontinuous Galerkin method for the Allen-Cahn equation are derived.Finally,five numerical examples are presented to verify the performance of numerical experiments.