Schur-complement-enhanced iterative solvers for coupled problems and phase field fracture models

• This study proposes a robust and efficient algorithm based on Schur complement to boost the solution of coupled problems. • The scheme is second-order convergent when the degree of coupling (DOC) is less than one. • For biconvex optimization problems, the scheme involves a modification when the DOC is not less than one. • The modification functions well also for the phase field approach to fracture and thermal cracking. • The proposed scheme can be generalized to multi-field coupled problems. Multi-field coupled problems are ubiquitous across many domains. We propose an efficient solution scheme based on the Schur complement for moderately and strongly coupled problems named Schur-complement-enhanced Newton iterative computing (SCENIC). For moderately coupled problems, SCENIC employs the Schur complement to update one field (the master field) while accounting for the influence of the other field (the slave field). After the master field is updated, the slave field is solved with any standard method (e.g., Newton’s method) while holding the master field fixed. This scheme is shown to be second-order convergent in this case. For a class of strongly coupled problems including biconvex optimization problems and the phase field approach to fracture, we introduce a modification to SCENIC to ensure its robustness. Compared with commonly used methods such as the staggered scheme and the monolithic scheme, SCENIC generally requires substantially fewer iterations and less CPU time.