Continuous and discontinuous Galerkin time stepping methods for nonlinear initial value problems with application to finite time blow-up

Abstract

We consider continuous and discontinuous Galerkin time stepping methods of arbitrary order as applied to nonlinear initial value problems in real Hilbert spaces. Our only assumption is that the nonlinearities are continuous; in particular, we include the case of unbounded nonlinear operators. Specifically, we develop new techniques to prove general Peano-type existence results for discrete solutions. In particular, our results show that the existence of solutions is independent of the local approximation order, and only requires the local time steps to be sufficiently small (independent of the polynomial degree). The uniqueness of (local) solutions is addressed as well. In addition, our theory is applied to finite time blow-up problems with nonlinearities of algebraic growth. For such problems we develop a time step selection algorithm for the purpose of numerically computing the blow-up time, and provide a convergence result.This effect is commonly termed (finite-time) blow-up.Galerkin Time Stepping. Galerkin-type time stepping methods for initial-value problems are based on weak formulations. For both the cG and the dG time stepping schemes, the test spaces consist of polynomials that are discontinuous at the time nodes. In this way, the discrete Galerkin formulations decouple into local problems on each time step, and the discretizations can therefore be understood as implicit one-step schemes. Galerkin time stepping methods have been analyzed for ordinary differential equations (ODEs), e.g., in [3,[5][6][7][8]10].A key feature of Galerkin time stepping methods is their great flexibility with respect to the size of the time steps and the local approximation orders, thereby naturally leading to an hp-version Galerkin framework. The hp-versions of the cG and dG time stepping schemes were introduced and analyzed in the works [12,13,15,19]. In particular, in the articles [12,19], which focus on ordinary initial value problems with uniform Lipschitz nonlinearities, the use of the contraction mapping theorem made it possible to prove existence and uniqueness results for discrete Galerkin solutions, which are independent of the local approximation orders. We emphasize that the hp-approach is well-known for its ability to approximate smooth solutions with possible local singularities at high algebraic or even exponential rates of convergence; see, e.g., [4,13,14,18] for the numerical approximation of problems with start-up singularities.

A posteriori error bounds for fully-discrete hp-discontinuous Galerkin timestepping methods for parabolic problems

“…for all t ∈ I n and W ∈ Y ; we refer to [35,Lemmas 6 & 7] or [20,Remark 1] for an explicit formula for κ n . Then, we can conclude…”

A note on a norm-preserving continuous Galerkin time stepping scheme

“…Suppose that the operator F satisfies (F1) and (F3), and that the exact solution u of (1) fulfills (11). Then, if U cG ∈ V r cG (M) solves the cG scheme (4)- (5), the a posteriori error bound…”

A Posteriori Error Analysis for Implicit–Explicit hp-Discontinuous Galerkin Timestepping Methods for Semilinear Parabolic Problems

“…This is in contrast with the increasing number of interesting works on a posteriori error analyses for low order time-stepping schemes for nonlinear evolution problems; see, e.g., [8,10,12,17,36,52] and the references therein. At the same time, there exist only few works on the a posteriori error analysis of high order time-stepping schemes for time-discrete nonlinear parabolic problems [30,33,34,41,46].…”

“…In this work, we use energy arguments for the proof a posteriori error bounds, as Sobolev imbeddings are sufficient for control of the nonlinear terms. For the case of blow-up problems an alternative technique based on Duhamel's principle combined with L ∞ -control bootstrapping arguments in the time variable is also available; we refer to [30,33,34] for time-discrete results in this vein.…”