Convergence Analysis of Spatially Adaptive Rothe Methods

On the Convergence Analysis of the Inexact Linearly Implicit Euler Scheme for a Class of Stochastic Partial Differential Equations

“…Therein, Rothe's method for deterministic parabolic equations is analyzed by putting it into an abstract framework, cf. in particular [7,Section 2]. We proceed similarly.…”

“…Proof. We argue following the lines of the proof of [7,Theorem 2.16]. To this end, we introduce the operators…”

“…Even for the deterministic case, not many results concerning a rigorous convergence and complexity analysis of the overall scheme seem to be available. To our best knowledge, the most far reaching achievements have been obtained in [7]. In this paper, it has been clarified how the tolerances for the elliptic subproblems in each time step have to be tuned so that the overall (perturbed) discretization scheme realizes the same order of convergence (in time direction) as the unperturbed one.…”

“…Moreover, based on concepts from approximation theory and function space theory, respectively, a complexity analysis of the overall scheme has been derived. It is the aim of this paper to generalize the analysis presented in [7] to SPDEs of the form (1). We mainly consider the case of the implicit Euler scheme, and we concentrate on the convergence analysis.…”

“…What we have in mind are applications of adaptive wavelet solvers, which are guaranteed to converge with optimal order, as developed, e.g., in [9], combined with efficient evaluations of the nonlinearities f as they can be found, e.g., in [11,16,30]. In [7] we have investigated how the error propagates within the linearly implicit Euler scheme and how the tolerances ε k in each time step have to be chosen, such that we obtain the same order of convergence as in the case of exact evaluation of the elliptic problems. We have shown that the tolerances depend on the Lipschitz constants C Lip τ,j,k of the operators…”