Abstract. This paper is concerned with the design and implementation of efficient solution algorithms for elliptic PDE problems with correlated random data. The energy orthogonality that is built into stochastic Galerkin approximations is cleverly exploited to give an innovative energy error estimation strategy that utilizes the tensor product structure of the approximation space. An associated error estimator is constructed and shown theoretically and numerically to be an effective mechanism for driving an adaptive refinement process. The codes used in the numerical studies are available online.Key words. stochastic Galerkin methods, stochastic finite elements, PDEs with random data, error estimation, a posteriori error analysis, adaptive methods, parametric operator equations AMS subject classifications. 35R60, 65C20, 65N30, 65N15.1. Introduction. Stochastic Galerkin approximation methods have emerged over the last decade as an efficient alternative to sampling methods for computing solutions (and associated quantities of interest) when studying linear 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. [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.An outline of the paper is as follows. Sections 2 and 3 set the problem of interest within the general framework of parametric operator equations with a potentially infinite-dimensional parameter space. The new error estimator is identified in Section 4. The estimator is shown to be reliable and efficient, and its properties are established that prove useful when individual error components are used to drive adaptive refinement. A specific implementation of an adaptive refinement strategy is described in section 5, and a set of numerical experiments that illustrate the effectiveness of the strategy is presented in section 6. One notable feature is that our software implementation is not limited to the lowest-order conforming spatial approximationthis means that we can solve spatially-regular problems to high accuracy with just a few adaptive refinement steps.

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

We prove that for compactly perturbed elliptic problems, where the corresponding bilinear form satisfies a Gårding inequality, adaptive mesh-refinement is capable of overcoming the preasymptotic behavior and eventually leads to convergence with optimal algebraic rates. As an important consequence of our analysis, one does not have to deal with the a priori assumption that the underlying meshes are sufficiently fine. Hence, the overall conclusion of our results is that adaptivity has stabilizing effects and can overcome possibly pessimistic restrictions on the meshes. In particular, our analysis covers adaptive mesh-refinement for the finite element discretization of the Helmholtz equation from where our interest originated.

We propose and analyze novel adaptive algorithms for the numerical solution of elliptic partial differential equations with parametric uncertainty. Four different marking strategies are employed for refinement of stochastic Galerkin finite element approximations. The algorithms are driven by the energy error reduction estimates derived from two-level a posteriori error indicators for spatial approximations and hierarchical a posteriori error indicators for parametric approximations. The focus of this work is on the mathematical foundation of the adaptive algorithms in the sense of rigorous convergence analysis. In particular, we prove that the proposed algorithms drive the underlying energy error estimates to zero.Date: November 26, 2018. 2010 Mathematics Subject Classification. 35R60, 65C20, 65N12, 65N15, 65N30. Key words and phrases. adaptive methods, a posteriori error analysis, convergence, two-level error estimate, stochastic Galerkin methods, finite element methods, parametric PDEs.Acknowledgements.

We prove an optimal a priori error estimate for the p-version of the boundary element method with hypersingular operators on piecewise plane open surfaces. The solutions of problems on open surfaces typically exhibit a singular behavior at the edges and corners of the surface which prevent an approximation analysis in H 1 . We analyze the approximation by polynomials of typical singular functions in fractional order Sobolev spaces thus giving, as an application, the optimal rate of convergence of the p-version of the boundary element method. This paper extends the results of [C. Schwab, M. Suri, The optimal p-version approximation of singularities on polyhedra in the boundary element method, SIAM J. Numer. Anal., 33 (1996), pp. 729-759] who only considered closed surfaces where the solution is in H 1 .