Stability and error estimates valid for infinite time, for strongly monotone and infinitely stiff evolution equations

Abstract

For evolution equation with a monotone operator we derive unconditional stability and error estimates valid for all times. θ-method, For the θ=1/2(2+ζτ^V), 0<v≤1, ζ>0), we prove an error estimate O(τ^4/3), τ → 0, if V=1/3, where τ is the maximal timestep for an arbitrary choice of the sequence of timesteps and with no further condition on F, and an estimate O(τ²) under some additional conditions. The first result is an improvement over the implicit midpoint method (θ=½), for which an order reduction to O(τ) may occur.