Energy Norm A Posteriori Error Estimation for Parametric Operator Equations

Abstract

Abstract. Stochastic Galerkin approximation is an increasingly popular approach for the solution of elliptic PDE problems with correlated random data. A typical strategy is to combine conventional (h-)finite element approximation on the spatial domain with spectral (p-)approximation on a finite-dimensional manifold in the (stochastic) parameter domain. The issues involved in a posteriori error analysis of computed solutions are outlined in this paper using an abstract setting of parametric operator equations. A novel energy error estimator that uses a parameter-free part of the underlying differential operator is introduced which effectively exploits the tensor product structure of the approximation space. We prove that our error estimator is reliable and efficient. We also discuss different strategies for enriching the approximation space and prove two-sided estimates of the error reduction for the corresponding enhanced approximations. These give computable estimates of the error reduction that depend only on the problem data and the original approximation.Key words. stochastic Galerkin methods, stochastic finite elements, random data, KarhunenLoève expansion, parametric operator equations, error estimation, a posteriori error analysis

Efficient Adaptive Stochastic Galerkin Methods for Parametric Operator Equations

“…The following result is an immediate consequence of Propositions 4.1 and 4.2 in [4]. It establishes the relation between the true error e in (4.1) and the estimator e * 0 satisfying (4.2).…”

“…The development of good/optimal adaptive refinement strategies remains an open question however. It is highlighted in our previous work [4] as well as by other researchers: notably Le Maître and collaborators [13], [14], [15], Wan & Karniadakis [19], [20], and Butler and collaborators [6], [5]. The strategy that is developed herein is similar in spirit to that developed by Eigel et al [7], but it is novel in that a posteriori estimates of the error reduction in the energy norm (rather than the error itself) are used to guide the adaptivity process.…”

“…holds with the same constant γ ∈ [0, 1) as in the strengthened Cauchy-Schwarz inequality (3.7) for the subspaces X, Y ⊂ H. This fact is due to the orthonormality of the polynomials in P P (see [4,Lemma 3.1] for the proof of (3.11)). We will define the enriched finite-dimensional subspace of V as the space…”

“…Following [4] we can approximate the error e ∈ V by using the bilinear form B 0 (·, ·) given by (2.6) instead of B(·, ·) on the left-hand side of (4.1), and by discretizing the resulting identity via Galerkin projection onto the enriched subspace V * XP given by (3.12). This leads to the error estimator e * 0 ∈ V * XP satisfying…”

“…The spatial error estimator e Y P is computed using the strategy described in the precursor paper [4]. On each element K ∈ ∆ h we use a standard element residual technique (see, e.g., Ainsworth & Oden [1]) to construct the following local residual problem denotes the flux jump in the approximate solution u XP across inter-element edges.…”