The derived-vector space framework and four general purposes massively parallel DDM algorithms

Ideally, DDMs seek what we call the DDM-paradigm: “constructing the global solution by solving local problems, exclusively”. To achieve it, it is essential to disconnect the subdomain-problems. In FETI-DP such disconnection is achieved by formulating the method in a product function-space that contains discontinuous functions. However, FETI-DP uses an indirect formulation based on Lagrange-multipliers. BDDC uses instead a more direct formulation, but does not work directly in a space of discontinuous functions, either. Another fact difficult to overcome is: at present competitive algorithms need to incorporate constraints that prevent full disconnection of the subdomains. This paper is devoted to explain a direct (primal) approach to DDMs in which all the numerical work is done in a product-space (the derived-vector space), which supplies a unified setting for non-overlapping DDMs and can be used to formulate and discuss in a general and systematic manner the theory of DDMs for non-symmetric problems. Furthermore, in this realm four general-purposes preconditioned algorithms with constraints applicable to non-symmetric matrices, which achieve the DDM-paradigm, have been obtained. Two of them have been identified as DVS-versions of BDDC and FETI-DP. The uniformity of the matrix-formulas expressing such algorithms should be highlighted.