ANALYSIS OF A FULLY DISCRETE FINITE ELEMENT METHOD FOR PARABOLIC INTERFACE PROBLEMS WITH NONSMOOTH INITIAL DATA

This article concerns numerical approximation of a parabolic interface problem with general L-2 initial value. The problem is discretized by a finite element method with a quasi-uniform triangulation of the domain fitting the interface, with piecewise linear approximation to the interface. The semi-discrete finite element problem is furthermore discretized in time by the k-step backward difference formula with k = 1, ...,6. To maintain high-order convergence in time for possibly nonsmooth L-2 initial value, we modify the standard backward difference formula at the first k-1 time levels by using a method recently developed for fractional evolution equations. An error bound of O(t(n)(-k)tau(k)+ t(n)(-1)h(2)vertical bar log h vertical bar) is established for the fully discrete finite element method for general L-2 initial data.