A posteriori error estimation and adaptive mesh refinement for parabolic interface problems using non-conforming immersed finite element method

Abstract

The purpose of this paper is to study a posteriori error analysis for parabolic interface problems using non-conforming immersed finite element method in a two-dimensional convex polygonal domain. The finite element discretization is such that mesh points need not fit the interface. While the piecewise linear finite elements are employed to approximate the spatial variable, the backward Euler method is used for the time discretization. The basic idea of the immersed finite element method is to modify the basis functions which satisfy the natural jump conditions across the interface. Some new error indicators are introduced to control the error due to non-body fitted mesh. For the adaptive mesh refinement procedure, the residual-based a posteriori error estimates are derived using the energy argument. A global upper and a local lower bounds for the error are established. A space–time adaptive algorithm is provided for the proposed method. Numerical results are reported to illustrate the performance of the derived error indicators.