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"Babuška, Ivo"
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A Stochastic Collocation Method for Elliptic Partial Differential Equations with Random Input Data
This work proposes and analyzes a stochastic collocation method for solving elliptic partial differential equations with random coefficients and forcing terms. These input data are assumed to depend on a finite number of random variables. The method consists of a Galerkin approximation in space and a collocation in the zeros of suitable tensor product orthogonal polynomials (Gauss points) in the probability space, and naturally leads to the solution of uncoupled deterministic problems as in the Monte Carlo approach. It treats easily a wide range of situations, such as input data that depend nonlinearly on the random variables, diffusivity coefficients with unbounded second moments, and random variables that are correlated or even unbounded. We provide a rigorous convergence analysis and demonstrate exponential convergence of the \"probability error\" with respect to the number of Gauss points in each direction of the probability space, under some regularity assumptions on the random input data. Numerical examples show the effectiveness of the method. Finally, we include a section with developments posterior to the original publication of this work. There we review sparse grid stochastic collocation methods, which are effective collocation strategies for problems that depend on a moderately large number of random variables.
Journal Article
A Stochastic Collocation Method for Elliptic Partial Differential Equations with Random Input Data
In this paper we propose and analyze a stochastic collocation method to solve elliptic partial differential equations with random coefficients and forcing terms (input data of the model). The input data are assumed to depend on a finite number of random variables. The method consists in a Galerkin approximation in space and a collocation in the zeros of suitable tensor product orthogonal polynomials (Gauss points) in the probability space and naturally leads to the solution of uncoupled deterministic problems as in the Monte Carlo approach. It can be seen as a generalization of the stochastic Galerkin method proposed in [I. Babuška, R. Tempone, and G. E. Zouraris, SIAM J. Numer. Anal., 42 (2004), pp. 800-825] and allows one to treat easily a wider range of situations, such as input data that depend nonlinearly on the random variables, diffusivity coefficients with unbounded second moments, and random variables that are correlated or even unbounded. We provide a rigorous convergence analysis and demonstrate exponential convergence of the \"probability error\" with respect to the number of Gauss points in each direction in the probability space, under some regularity assumptions on the random input data. Numerical examples show the effectiveness of the method.
Journal Article
A RESIDUAL-BASED A POSTERIORI ERROR ESTIMATOR FOR THE STOKES-DARCY COUPLED PROBLEM
In this paper we develop an a posteriori error analysis of a new conforming mixed finite element method for the coupling of fluid flow with porous media flow. The flows are governed by the Stokes and Darcy equations, respectively, and the transmission conditions are given by mass conservation, balance of normal forces, and the Beavers-Joseph-Saffman law. The finite element subspaces consider Bernardi-Raugel and Raviart-Thomas elements for the velocities, piecewise constants for the pressures, and continuous piecewise linear elements for a Lagrange multiplier defined on the interface. We derive a reliable and efficient residual-based a posteriori error estimator for this coupled problem. The proof of reliability makes use of suitable auxiliary problems, diverse continuous inf-sup conditions satisfied by the bilinear forms involved, and local approximation properties of the Clément interpolant and Raviart-Thomas operator. On the other hand, Helmholtz decomposition, inverse inequalities, and the localization technique based on triangle-bubble and edge-bubble functions are the main tools for proving the efficiency of the estimator. Up to minor modifications, our analysis can be extended to other finite element subspaces yielding a stable Galerkin scheme.
Journal Article
Finite element analysis : method, verification and validation
Finite Element Analysis An updated and comprehensive review of the theoretical foundation of the finite element method The revised and updated second edition of Finite Element Analysis: Method, Verification, and Validation offers a comprehensive review of the theoretical foundations of the finite element method and highlights the fundamentals of.
eBook
Introduction to Finite Element Analysis
When using numerical simulation to make a decision, how can its reliability be determined? What are the common pitfalls and mistakes when assessing the trustworthiness of computed information, and how can they be avoided? Whenever numerical simulation is employed in connection with engineering decision-making, there is an implied expectation of reliability: one cannot base decisions on computed information without believing that information is reliable enough to support those decisions. Using mathematical models to show the reliability of computer-generated information is an essential part of any modelling effort. Giving users of finite element analysis (FEA) software an introduction to verification and validation procedures, this book thoroughly covers the fundamentals of assuring reliability in numerical simulation. The renowned authors systematically guide readers through the basic theory and algorithmic structure of the finite element method, using helpful examples and exercises throughout. * Delivers the tools needed to have a working knowledge of the finite element method * Illustrates the concepts and procedures of verification and validation * Explains the process of conceptualization supported by virtual experimentation * Describes the convergence characteristics of the h-, p- and hp-methods * Covers the hierarchic view of mathematical models and finite element spaces * Uses examples and exercises which illustrate the techniques and procedures of quality assurance * Ideal for mechanical and structural engineering students, practicing engineers and applied mathematicians * Includes parameter-controlled examples of solved problems in a companion website (www.wiley.com/go/szabo [http://www.wiley.com/go/szabo])
eBook
Finite Element Solution of the Helmholtz Equation with High Wave Number Part II: The h-p Version of the FEM
In this paper, which is part II in a series of two, the investigation of the Galerkin finite element solution to the Helmholtz equation is continued. While part I contained results on the h version with piecewise linear approximation, the present part deals with approximation spaces of order p ≥ 1. As in part I, the results are presented on a one-dimensional model problem with Dirichlet-Robin boundary conditions. In particular, there are proven stability estimates, both with respect to data of higher regularity and data that is bounded in lower norms. The estimates are shown both for the continuous and the discrete spaces under consideration. Further, there is proven a result on the phase difference between the exact and the Galerkin finite element solutions for arbitrary p that had been previously conjectured from numerical experiments. These results and further preparatory statements are then employed to show error estimates for the Galerkin finite element method (FEM). It becomes evident that the error estimate for higher approximation can--with certain assumptions on the data--be written in the same form as the piecewise linear case, namely, as the sum of the error of best approximation plus a pollution term that is of the order of the phase difference. The paper is concluded with a numerical evaluation.
Journal Article
LOCAL JACOBI OPERATORS AND APPLICATIONS TO THE p-VERSION OF FINITE ELEMENT METHOD IN TWO DIMENSIONS
Based on the Chebyshev projection-interpolation on each edge of elements and the Chebyshev projection on each element, we have designed the local Jacobi operators ΠΩ j on the triangular or quadrilateral element Ω j , 1 < j < J such that ΠΩ j u is a polynomial of degree p on Ω j which interpolates u at the vertices of Ω j , coincides with the Chebyshev projection-interpolation of u on the edges of Ω j , and possesses the best approximation to the smooth and singular functions u. By a simple assembly of ΠΩ j u, 1 < j < J we construct a piecewise and globally continuous polynomial of degree p which has the best approximation error bound locally and globally for singular as well as smooth solutions on general quasi-uniform meshes and satisfies the homogeneous Dirichlet boundary conditions. An application of the local Jacobi operators to the p-version of the finite element method associated with general meshes composed of (curvilinear) triangular and quadrilateral elements for problems on polygonal domains leads to the optimal convergence, which has been an open problem for more than three decades.
Journal Article
Mechanics of Materials with Periodic Truss or Frame Micro-Structures
This paper describes the mechanics of materials with periodic skeletal micro-structures in infinite domains. The principal technical results consist of certain Korn-type inequalities that provide upper and lower bounds for the linear elastic strain energy in the material. Using these inequalities, existence and uniqueness results for the equations of linear elastic equilibrium are derived, and some asymptotic properties of the solutions are described. Particular attention is paid to the question of when a lattice structure can accurately be modeled as a pin-jointed truss, and when a rigid-node frame model must be employed. A practical technique for how to distinguish between the two types of material is given, and the distinct differences in their mechanical behavior are described.[PUBLICATION ABSTRACT]
Journal Article
Direct and Inverse Approximation Theorems for the p-Version of the Finite Element Method in the Framework of Weighted Besov Spaces. Part I: Approximability of Functions in the Weighted Besov Spaces
This is the first of a series devoted to the approximation theory of the p-version of the finite element method in two dimensions in the framework of the Jacobi-weighted Besov spaces, which provides the p-version with a solid mathematical foundation. In this paper, we establish a mathematical framework of the Jacobi-weighted Besov and Sobolev spaces and analyze the approximability of the functions in the framework of these spaces, particularly, singular functions of$r^{\\gamma}$-type and$r^{\\gamma} \\log^{\\nu}$r-type. These spaces and the corresponding approximation properties are of fundamental importance to the proof of the optimal convergence for the p-version in two dimensions in part II and to various sharp inverse approximation theorems in part III.
Journal Article