A partially sequential preconditioner for a parallel and efficent finite element solution

A coarse/fine mesh preconditioner is presented which can be successfully applied in a parallel finite element context. The proposed method involves the reconstruction of the stiffness equations using a coarse/fine mesh idealisation with relative degrees-of-freedom derived directly from the element shape functions. This approach leads naturally to an effective preconditioner which only requires a direct solution on coarse mesh variables and which is implemented sequentially. On the other hand, the preconditioning of the fine mesh variables involves a perfectly parallelizable diagonal scaling. The proposed derivation of the coarse/fine mesh discretization via the use of transformation matrices can be very efficient and is directly applicable to existing finite elememt solution procedures.