THE PARTITION OF UNITY METHOD

A new Þnite element method is presented that features the ability to include in the Þnite element space knowledge about the partial di¤erential equation being solved. This new method can therefore be more e¦cient than the usual Þnite element methods. An additional feature of the partition-of-unity method is that Þnite element spaces of any desired regularity can be constructed very easily. This paper includes a convergence proof of this method and illustrates its e¦ciency 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.