Ivo Babuška

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 the differential equation under consideration to construct effective and robust methods. Although the method and its theory are valid in n dimensions, a detailed and illustrative analysis will be given for a one-dimensional model problem. We identify some classes of non-standard problems which can profit highly from the advantages of the PUFEM and conclude this paper with some open questions concerning implementational aspects of the PUFEM.

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 therefore be more efficient than the usual finite element methods. An additional feature of the partition-of-unity method is that finite element spaces of any desired regularity can be constructed very easily. This paper includes a convergence proof of this method and illustrates its efficiency by an application to the Helmholtz equation for high wave numbers. The basic estimates for a posteriori error estimation for this new method are also proved.

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 essential (stable) boundary conditions. The implementation is based on the application of Lagrangian multiplier. The rate of convergence is proved.

The p-Version of the Finite Element Method

Abstract : The finite element method has become the main tool in computational mechanics. The MAKABASE contains about 20,000 references on finite element and 2000 boundary element technology. Recently the new direction in the finite element theory and practice appeared, the p and h-p versions, which utilize high degree of elements. About 3 - 4 dozen references about p and h-p versions are available, all of them related to the elliptic problems. For the survey of todays state of the art, we refer to e.g. This paper addresses the basic problems of the p-version for the parabolic equation with both variables, x and t discreted via p-version. It concentrates on the case when in the time variables only one interval is used. The paper gives the error estimates and presents some numerical aspects. The authors restrict themselves to the basic features of the method. Various generalizations will be presented in forthcoming papers.

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 standard finite element spaces are augmented by adding special functions which reflect the known information about the boundary value problem and the input data; e.g., the singular functions obtained from the local asymptotic expansion of the exact solution in the neighborhood of a corner point, etc. The special functions are multiplied with the partition of unity corresponding to the standard linear vertex shape-functions and pasted together with the existing finite element basis to construct an augmented conforming finite element space. In this way, the local approximability afforded by the special functions is included in the approximation, while maintaining the existing infrastructure of finite element codes. The major features of the GFEM are: (1) the essential boundary conditions can be imposed exactly as in the standard FEM, unlike other partition of unity based methods where this is a major issue; (2) the accuracy of the numerical integration of the entries of the stiffness matrix and load vector is controlled adaptively so that the errors in integration of the special functions do not affect the accuracy of the constructed approximation (this issue also has not been sufficiently addressed in other implementations of partition of unity or meshless methods); and (3) linear dependencies in the system of equations are resolved by employing an easy modification of the direct linear solver. The power of the GFEM for solving problems in domains with complex geometry with less error and less computer resources than the standard FEM is illustrated by numerical examples.

Galerkin Finite Element Approximations of Stochastic Elliptic Partial Differential Equations

We describe and analyze two numerical methods for a linear elliptic problem with stochastic coefficients and homogeneous Dirichlet boundary conditions. Here the aim of the com- putations is to approximate statistical moments of the solution, and, in particular, we give a priori error estimates for the computation of the expected value of the solution. The first method gener- ates independent identically distributed approximations of the solution by sampling the coefficients of the equation and using a standard Galerkin finite element variational formulation. The Monte Carlo method then uses these approximations to compute corresponding sample averages. The sec- ond method is based on a finite dimensional approximation of the stochastic coefficients, turning the original stochastic problem into a deterministic parametric elliptic problem. A Galerkin finite element method, of either the h -o rp-version, then approximates the corresponding deterministic solution, yielding approximations of the desired statistics. We present a priori error estimates and include a comparison of the computational work required by each numerical approximation to achieve a given accuracy. This comparison suggests intuitive conditions for an optimal selection of the numerical approximation.

ON THE ANGLE CONDITION IN THE FINITE ELEMENT METHOD

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 frequently used approaches is to triangulate the domain and find the approximate solution which is linear, quadratic, etc., in every triangle. A condition which is considered essential is that the angle of every triangle, independent of its size, should not be small. In this paper it is shown that the minimum angle condition is not essential. What is essential is the fact that no angle is too close to

The generalized finite element method

180^ \circ

Special finite element methods for a class of second order elliptic problems with rough coefficients

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Generalized finite element methods for three-dimensional structural mechanics problems

This paper describes a pilot design and implementation of the generalized finite element method (GFEM), as a direct extension of the standard finite element method (SFEM, or FEM), which makes possible the accurate solution of engineering problems in complex domains which may be practically impossible to solve using the FEM. The development of the GFEM is illustrated for the Laplacian in two space dimensions in domains which may include several hundreds of voids, and/or cracks, for which the construction of meshes used by the FEM is practically impossible. The two main capabilities are: (1) It can construct the approximation using meshes which may overlap part, or all, of the domain boundary. (2) It can incorporate into the approximation handbook functions, which are known analytically, or are generated numerically, and approximate well the solution of the boundary value problem in the neighborhood of corner points, voids, cracks, etc. The main tool is a special integration algorithm, which we call the Fast Remeshing approach, which is robust and works for any domain with arbitrary complexity. The incorporation of the handbook functions into the GFEM is done by employing the partition of unity method (PUM). The presented formulations and implementations can be easily extended to the multi-material medium where the voids are replaced by inclusions of various shapes and sizes, and to the case of the elasticity problem. This work can also be understood as a pilot study for the feasibility and demonstration of the capabilities of the GFEM, which is needed before analogous implementations are attempted in the three-dmensional and nonlinear cases, which are the cases of main interest for future work.