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The partition of unity finite element method: Basic theory and applications
The paper presents the basic ideas and the mathematical foundation of the partition of unity finite element method (PUFEM). We will show how the PUFEM can be used to employ the structure of theExpand
The Partition of Unity Method
A new finite element method is presented that features the ability to include in the finite element space knowledge about the partial differential equation being solved. This new method can thereforeExpand
A Stochastic Collocation Method for Elliptic Partial Differential Equations with Random Input Data
A rigorous convergence analysis is provided and exponential convergence of the “probability error” with respect to the number of Gauss points in each direction in the probability space is demonstrated, under some regularity assumptions on the random input data. Expand
The finite element method with Lagrangian multipliers
SummaryThe Dirichlet problem for second order differential equations is chosen as a model problem to show how the finite element method may be implemented to avoid difficulty in fulfilling essentialExpand
Galerkin Finite Element Approximations of Stochastic Elliptic Partial Differential Equations
A priori error estimates for the computation of the expected value of the solution are given and a comparison of the computational work required by each numerical approximation is included to suggest intuitive conditions for an optimal selection of the numerical approximation. Expand
The design and analysis of the Generalized Finite Element Method
Abstract In this paper, we introduce the Generalized Finite Element Method (GFEM) as a combination of the classical Finite Element Method (FEM) and the Partition of Unity Method (PUM). The standardExpand
The finite element procedure consists in finding an approximate solution in the form of piecewise linear functions, piecewise quadratic, etc. For two-dimensional problems, one of the most frequentlyExpand
The finite element method and its reliability
Preface 1. Introduction 2. Mathematical formulation of the model problem 3. The finite element method 4. Local behaviour in the finite element method 5. A-posteriori estimation of the error 6.Expand