Analysis of backward Euler/extended finite element discretization of parabolic problems with moving interfaces

Numerical experiments performed by several authors repeatedly confirmed that the extended finite element method combined with classical time stepping schemes, such as backward Euler or Crank–Nicolson, provides optimal rates of convergence when applied to problems with moving interfaces. In spite of the rapid spread of this discretization approach, a complete space and time error analysis is not available yet. The objective of this work is to contribute to closing the gap between observations and rigorous analysis. More precisely, we preform a thorough space and time error analysis of different variants of the extended finite element method combined with backward Euler time advancing scheme when applied to problems with moving interfaces. Our results prove that the discretization error, measured in the natural norms, decreases with the expected rate, when space and time discretization steps are refined. We also show that the space and time approximation properties are fully decoupled. As a consequence, high order approximation schemes could be developed and analyzed within the theoretical framework proposed here. Numerical experiments are finally addressed for the verification of the algorithms. Besides the interest of providing rigorous error bounds, we believe that a general theoretical framework is extremely helpful as a guide for further developing and refining extended finite element methods for free interface problems.