The paper presents the basic ideas and the mathematical foundation of the partition of unity nite element method (PUFEM). We will show h o w the PUFEM can be used to employ the structure of the di erential equation under consideration to construct e ective 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 pro t highly from the advantages of the PUFEM and conclude this paper with some open questions concerning implementational aspects of the PUFEM.Keywords: Finite element method, meshless nite element method, robust nite element methods, nite element methods for highly oscillatory solutions

Summary. The 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.

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 computations 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 generates 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 second 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-or p-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.

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.KEY WORDS: finite element method; meshless finite element method; finite element methods for highly oscillatory solutions