The Partition of Unity Finite Element Method: Basic Theory and Applications

Abstract : 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 finite element method is that finite element spaces of any desired regularity can be constructed very easily. Moreover the method is of meshless type. 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. (AN)