Galerkin time-stepping methods for nonlinear parabolic equations

Abstract

Abstract.We consider discontinuous as well as continuous Galerkin methods for the time discretization of a class of nonlinear parabolic equations. We show existence and local uniqueness and derive optimal order optimal regularity a priori error estimates. We establish the results in an abstract Hilbert space setting and apply them to a quasilinear parabolic equation.Mathematics Subject Classification. 65M15, 65M50.

Optimal order a posteriori error estimates for a class of Runge–Kutta and Galerkin methods

“…. , N. For local uniqueness and existence results for cG as well as for a priori error estimates, including nonlinear parabolic equations, we refer to [1,4]. It follows from (3.1) that U ∈ V q satisfies also the following pointwise equation…”

Convergence of discontinuous Galerkin approximations of an optimal control problem associated to semilinear parabolic PDE's

“…The relation of the DG method to adaptive techniques was studied in [11,12,39]. Results related to finite element approximation of semi-linear and general nonlinear parabolic problems are presented in [1,[13][14][15].…”

“…For k > 1, existence and uniqueness can be proved under local Lipschitz continuity properties for more general nonlinear problems by using fixed point arguments (see, e.g. [1] and references within).…”

Error estimates for a FitzHugh–Nagumo parameter-dependent reaction-diffusion system

“…The existence and uniqueness of discontinuous Galerkin approximations can be proved easily in case k = 0, 1. For the case k > 1, the existence and (local) uniqueness can be proved around the continuous (smooth) solution u, v (in an appropriate "parabolic" cube), provided that the semi-linear term satisfies suitable continuity and monotonicity assumptions which allow the application of standard fixed point theorems (see, e.g., [2,15,40]). Existence of discrete schemes of arbitrary order k under minimal regularity assumptions on the data can be proved analogously via fixed point theorems, while uniqueness follows by standard arguments upon deriving stability estimates (see the subsequent section).…”

“…An imlpicit-explicit multistep method for approximations of semi-linear parabolic PDEs is analyzed in the work of [3] and while linear implicit schemes were studied in [1]. The discontinuous (in time) Galerkin technique is analyzed in the works of [2,9,[11][12][13][14][15][30][31][32] for linear and semilinear problems. Several results regarding a posteriori error estimation of reaction-diffusion systems are presented in [16] (see also references within), while space-time adaptivity techniques for reaction-diffusion type of systems (including examples for the FHN system) are proposed in [20].…”