Stability and error estimates for the $\theta$-method for strongly monotone and infinitely stiff evolution equations

Summary.   For evolution equations with a strongly monotone operator we derive unconditional stability and discretization error estimates valid for all . For the -method, with , we prove an error estimate , if , where is the maximal integration step for an arbitrary choice of sequence of steps and with no assumptions about the existence of the Jacobian as well as other derivatives of the operator , and an optimal estimate under some additional relation between neighboring steps. The first result is an improvement over the implicit midpoint method , for which an order reduction to sometimes may occur for infinitely stiff problems. Numerical tests illustrate the results. Received March 10, 1999 / Revised version received April 3, 2000 / Published online February 5, 2001