A brief overview of non-overlapping domain decomposition methods

Abstract

An overview of non–overlapping domain decomposition methods is presented. The most efficient methods that exist at present, BDDC and FETI–DP, are placed in a 'primal' framework (the 'derived–vectors space (DVS)') which permits a synthetic and effective presentation of both: primal and 'dual' formulations. The derived–vectors space is similar to the setting used in BDDC. A significant difference is that, in the DVS framework, the problem considered is transformed into one that is defined in a product vector space while in BDDC that is not done. This simplifies the algorithmic formulations, which are summarized in a set of matrix–formulas applicable to symmetric, non–symmetric and indefinite matrices generated when treating numerically partial differential equations or systems of such equations. They can directly be used for code development. Two preconditioned algorithms of the mentioned set had not been reported previously in the DDM literature, as far as we know, and are suitable for being researched.