A Posteriori Error Estimates For The Variable BDF₂ Method For Parabolic Equations

Parabolic equation is widely used in physics,chemistry and so on.Therefore,it is very important to study the numerical methods for parabolic equation.Many researchers consider the a posteriori error estimates for variable-step size parabolic equations such as Crank-Nicolson,Runge-Kutta and Galerkin one-step methods.As for multi-step methods,a posteriori error estimates for BDF fixed step size has been studied in the literature.Since the fixed step-size has a great deal of limitation in the actual calculation,variable step-size is the basis of error control and adaptive computation,it is necessary to study the a posteriori error estimates.In this paper,we consider the following parabolic equations u'(t)+ Au(t)= f(t),0 ? t ? T,u(0)= u0.and investigate a posteriori error estimates for second variable step-size BDF2 method.1.We obtain the upper bounds and lower bounds of e = u-U and e = u-U by considering the approximations U obtained by second variable step size BDF2 and the reconstruction U obtained by the quadratic interpolation.2.We obtain the optimal order of posteriori error estimate.3.The results are extended to nonlinear parabolic equations.4.The effect of the first step with trapezoidal method are confirmed by numerical examples.