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.
Where a licence is displayed above, please note the terms and conditions of the licence govern your use of this document. When citing, please reference the published version. Take down policy While the University of Birmingham exercises care and attention in making items available there are rare occasions when an item has been uploaded in error or has been deemed to be commercially or otherwise sensitive.
We use the ideas of goal-oriented error estimation and adaptivity to design and implement an efficient adaptive algorithm for approximating linear quantities of interest derived from solutions to elliptic partial differential equations (PDEs) with parametric or uncertain inputs. In the algorithm, the stochastic Galerkin finite element method (sGFEM) is used to approximate the solutions to primal and dual problems that depend on a countably infinite number of uncertain parameters. Adaptive refinement is guided by an innovative strategy that combines the error reduction indicators computed for spatial and parametric components of the primal and dual solutions. The key theoretical ingredient is a novel two-level a posteriori estimate of the energy error in sGFEM approximations. We prove that this error estimate is reliable and efficient. The effectiveness of the goal-oriented error estimation strategy and the performance of the goal-oriented adaptive algorithm are tested numerically for three representative model problems with parametric coefficients and for three quantities of interest (including the approximation of pointwise values).Adaptive techniques based on rigorous a posteriori error analysis of computed solutions provide an effective mechanism for building approximation spaces and accelerating convergence of computed solutions. These techniques rely heavily on how the approximation error is estimated and controlled. One may choose to estimate the error in the global energy norm and use the associated error indicators to enhance the computed solution and drive the energy error estimate to zero. However, in practical applications, simulations often target a specific (e.g., localized) feature of the solution, called the quantity of interest and represented using a linear functional of the solution. In these cases, the energy norm may give very little useful information about the simulation error.Alternative error estimation techniques, such as goal-oriented error estimations, e.g., by the dual-weighted residual methods, allow to control the errors in the quantity of interest. While for deterministic PDEs, these error estimation techniques and the associated adaptive algorithms are very well studied (see, e.g., [EEHJ95, JS95, BR96, PO99, BR01, GS02, BR03] for the a posteriori error estimation and [MS09, BET11, HP16, FPZ16] for a rigorous convergence analysis of adaptive algorithms), relatively little work has been done for PDEs with parametric or uncertain inputs. For example, in the framework of (intrusive) stochastic Galerkin finite element methods (sGFEMs) (see, e.g., [GS91, LPS14]), a posteriori error estimation of linear functionals of solutions to PDEs with parametric uncertainty is addressed in [MLM07] and, for nonlinear problems, in [BDW11]. In particular, in [MLM07], a rigorous estimator for the error in the quantity of interest is derived and several adaptive refinement strategies are discussed. However, the authors comment that the proposed estimator lacks information about the structure of the estim...
3D printers based on the additive manufacturing technology create objects layer-by-layer dropping fused material. As a consequence, strong overhangs cannot be printed because the new-come material does not find a suitable support over the last deposed layer. In these cases, one can add support structures (scaffolds) which make the object printable, to be removed at the end. In this paper we propose a level set based method to create object-dependent support structures, specifically conceived to reduce both the amount of additional material and the printing time. We also review some open problems about 3D printing which can be of interests for the mathematical community.
T-IFISS is a finite element software package for studying finite element solution algorithms for deterministic and parametric elliptic partial differential equations. The emphasis is on self-adaptive algorithms with rigorous error control using a variety of a posteriori error estimation techniques. The open-source MATLAB framework provides a computational laboratory for experimentation and exploration, enabling users to quickly develop new discretizations and test alternative algorithms. The package is also valuable as a teaching tool for students who want to learn about state-of-the-art finite element methodology. and is compatible with Windows, Linux and MacOS computers.3 We would almost certainly use an iterative solver preconditioned with an algebraic multigrid V-cycle if we were trying to solve the same PDE problem in three spatial dimensions. 4 The default uniform refinement in T-IFISS is by three bisections (see Figure 1(d)). However, there is an option to switch to the so-called red uniform refinement (i.e., the one obtained by connecting the edge midpoints of each triangle); this can be done by setting subdivPar = 1 within the function adiff adaptive main.m.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.
hi@scite.ai
10624 S. Eastern Ave., Ste. A-614
Henderson, NV 89052, USA
Copyright © 2024 scite LLC. All rights reserved.
Made with 💙 for researchers
Part of the Research Solutions Family.