Continuous and Discontinuous Galerkin Time Stepping Methods for Nonlinear Initial Value Problems with Application to Finite Time Blow-Up

摘要

We consider continuous and discontinuous Galerkin time stepping methods ofarbitrary order as applied to nonlinear initial value problems in real Hilbertspaces. Our only assumption is that the nonlinearities are continuous; inparticular, we include the case of unbounded nonlinear operators. Specifically,we develop new techniques to prove general Peano-type existence results fordiscrete solutions. In particular, our results show that the existence ofsolutions is independent of the local approximation order, and only requiresthe local time steps to be sufficiently small (independent of the polynomialdegree). The uniqueness of (local) solutions is addressed as well. In addition,our theory is applied to finite time blow-up problems with nonlinearities ofalgebraic growth. For such problems we develop a time step selection algorithmfor the purpose of numerically computing the blow-up time, and provide aconvergence result.