Residual-Based A Posteriori Error Estimation For Elliptic Interface Problems Approximated By Immersed Finite Element Methods

In this thesis,we consider a residual-based a posteriori error estimation for partially penalized immersed finite element(PPIFE)approximation to elliptic interface problems.In the framework of PPIFE theory,we first obtain a usual representation of the approximation error and then define an a posteriori error estimator consisting of an elemental residual,a numerical flux jump,and a numerical solution jump.The PPIFE method is based on non-body-fitted mesh,and hence we perform detailed analysis on the local e ciency bounds of the estimator on regular and irregular interface elements with different techniques.We introduce a new approach,which does not involve the Helmholtz decomposition,to give the upper bounds of the approximation error with an L~2representation of the true error as the main tool.More importantly,the e ciency and reliability constants are independent of the interface location and the mesh size.In fact,the IFE method alone may not be e cient in solving interface problems involving singularity or steep gradient.We propose an adaptive refinement strategy that adapts the grid according to the information provided by the a posteriori error,thereby obtaining a grid more suited to the interface problem.Numerical results show that the adaptive strategy can effectively capture the error distribution and automatically refine the mesh near the interface or singularity.Besides,the adaptive PPIFE method improves computational accuracy and can accurately solve elliptic interface prob-lems with singularity or a large coe cient jump ratio.