(untitled) (Journal of Mathematical Study, 2019)
We are interested in the design of parallel numerical schemes for linear systems. We give an effective solution to this problem in the following case: the matrix A of the linear system is the product of p nonsingular matrices A_i^m with specific shape: A_i= I -h_iX for a fixed matrix X and real numbers h_i. Although having a special form, these matrices A_i arise frequently in the discretization of evolutionary Partial Differential Equations. For example, one step of the implicit Euler scheme for the evolution equation u' =Xu reads (I-hX)u^n+1=u^n. Iterating m times such a scheme leads to a linear system Au^n+m=u^n. The idea is to express A^-1 as a linear combination of elementary matrices A_i^-1 (or more generally in term of matrices A_i^-k). Hence the solution of the linear system with matrix A is a linear combination of the solutions of linear systems with matrices A_i (or A_i^k). These systems are then solved simultaneously on different processors.