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

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

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In this paper, we develop a new stability and convergence theory for highly indefinite elliptic partial differential equations by considering the Helmholtz equation at high wave number as our model problem. The key element in this theory is a novel k-explicit regularity theory for Helmholtz boundary value problems that is based on decomposing the solution into in two parts: the first part has the H 2 -Sobolev regularity expected of elliptic PDEs but features k-independent regularity constants; the second part is an analytic function for which k-explicit bounds for all derivatives are given. This decomposition is worked out in detail for several types of boundary value problems including the case Robin boundary conditions in domains with analytic boundary and in convex polygons.As the most important practical application we apply our full error analysis to the classical hp-version of the finite element method (hp-FEM) where the dependence on the mesh width h, the approximation order p, and the wave number k is given explicitly. In particular, under the assumption that the solution operator for Helmholtz problems grows only polynomially in k, it is shown that quasi-optimality is obtained under the conditions that kh/p is sufficiently small and the polynomial degree p is at least O(log k).

Abstract. A rigorous convergence theory for Galerkin methods for a model Helmholtz problem in R d , d ∈ {1, 2, 3} is presented. General conditions on the approximation properties of the approximation space are stated that ensure quasi-optimality of the method. As an application of the general theory, a full error analysis of the classical hp-version of the finite element method (hp-FEM) is presented for the model problem where the dependence on the mesh width h, the approximation order p, and the wave number k is given explicitly. In particular, it is shown that quasi-optimality is obtained under the conditions that kh/p is sufficiently small and the polynomial degree p is at least O(log k).