A Posteriori Error Analysis of Two-Step Backward Differentiation Formula Finite Element Approximation for Parabolic Interface Problems

Abstract

This paper studies a residual-based a posteriori error estimates for linear parabolic interface problems in a bounded convex polygonal domain in $$\mathbb {R}^2$$R2. We use the standard linear finite element spaces in space which are allowed to change in time and the two-step backward differentiation formula (BDF-2) approximation at equidistant time step is used for the time discretizations. The essential ingredients in the error analysis are the continuous piecewise quadratic space---time BDF-2 reconstruction and Scott---Zhang interpolation estimates. Optimal order in time and an almost optimal order in space error estimates are derived in the $$L^{\infty }(L^{2})$$Lź(L2)-norm using only energy method. The interfaces are assumed to be of arbitrary shape but are smooth for our purpose. Numerical experiments are performed to validate the asymptotic behaviour of the derived error estimators.