The focus of this work is a posteriori error estimation for stochastic Galerkin approximations of parameter-dependent linear elasticity equations. The starting point is a three-field partial ...differential equation model with the Young modulus represented as an affine function of a countable set of parameters. We introduce a weak formulation, establish its stability with respect to a weighted norm and discuss its approximation using stochastic Galerkin mixed finite element methods. We motivate an a posteriori error estimation scheme and establish upper and lower bounds for the approximation error. The constants in the bounds are independent of the Poisson ratio as well as the spatial and parametric discretisation parameters. We also discuss proxies for the error reduction associated with enrichments of the approximation spaces and we develop an adaptive algorithm that terminates when the estimated error falls below a user-prescribed tolerance. The error reduction proxies are shown to be reliable and efficient in the incompressible limit case. Numerical results are presented to supplement the theory. All experiments were performed using open source (IFISS) software that is available online.
We consider a linear elliptic partial differential equation (PDE) with a generic uniformly bounded parametric coefficient. The solution to this PDE problem is approximated in the framework of ...stochastic Galerkin finite element methods. We perform a posteriori error analysis of Galerkin approximations and derive a reliable and efficient estimate for the energy error in these approximations. Practical versions of this error estimate are discussed and tested numerically for a model problem with non-affine parametric representation of the coefficient. Furthermore, we use the error reduction indicators derived from spatial and parametric error estimators to guide an adaptive solution algorithm for the given parametric PDE problem. The performance of the adaptive algorithm is tested numerically for model problems with two different non-affine parametric representations of the coefficient.
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GEOZS, IJS, IMTLJ, KILJ, KISLJ, NLZOH, NUK, OILJ, PNG, SAZU, SBCE, SBJE, UILJ, UL, UM, UPCLJ, UPUK, ZAGLJ, ZRSKP
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.
•Adaptive mesh-refinement stabilizes FEM computations for indefinite problems.•The analysis avoids any assumption on the initial mesh.•The algorithm guarantees optimal convergence rates in the sense of certain nonlinear convergence classes.•Numerical experiments for the 2D Helmholtz equation underpin the theoretical findings.
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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.
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Abstract
We analyze an adaptive algorithm for the numerical solution of parametric elliptic partial differential equations in two-dimensional physical domains, with coefficients and right-hand-side ...functions depending on infinitely many (stochastic) parameters. The algorithm generates multilevel stochastic Galerkin approximations; these are represented in terms of a sparse generalized polynomial chaos expansion with coefficients residing in finite element spaces associated with different locally refined meshes. Adaptivity is driven by a two-level a posteriori error estimator and employs a Dörfler-type marking on the joint set of spatial and parametric error indicators. We show that, under an appropriate saturation assumption, the proposed adaptive strategy yields optimal convergence rates with respect to the overall dimension of the underlying multilevel approximation spaces.
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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).
•Novel two-level a posteriori error estimates for stochastic Galerkin approximations.•Effective goal-oriented error estimation strategy for stochastic Galerkin FEM.•Efficient goal-oriented adaptive sGFEM algorithm driven by error reduction indicators.
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We analyze an adaptive boundary element method for the weakly-singular and hypersingular integral equations for the 2D and 3D Helmholtz problem. The proposed adaptive algorithm is steered by a ...residual error estimator and does not rely on any a priori information that the underlying meshes are sufficiently fine. We prove convergence of the error estimator with optimal algebraic rates, independently of the (coarse) initial mesh. As a technical contribution, we prove certain local inverse-type estimates for the boundary integral operators associated with the Helmholtz equation.
•Adaptive mesh-refinement stabilizes BEM computations for indefinite problems.•Local inverse-type estimates for the integral operators of the Helmholtz equation.•The analysis avoids any assumption on the initial mesh.•The algorithm guarantees optimal convergence rates.•Numerical experiments for 3D acoustic scattering problems.
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