A dual mesh multigrid preconditioner for the efficient solution of hydraulically driven fracture problems

We present a novel multigrid (MG) procedure for the efficient solution of the large non-symmetric system of algebraic equations used to model the evolution of a hydraulically driven fracture in a multi-layered elastic medium. The governing equations involve a highly non-linear coupled system of integro-partial differential equations along with the fracture front free boundary problem. The conditioning of the algebraic equations typically degrades as O(N3). A number of characteristics of this problem present significant new challenges for designing an effective MG strategy. Large changes in the coefficients of the PDE are dealt with by taking the appropriate harmonic averages of the discrete coefficients. Coarse level Green’s functions for multiple elastic layers are constructed using a single dual mesh and superposition. Coarse grids that are sub-sets of the finest grid are used to treat mixed variable problems associated with ‘pinch points.’ Localized approximations to the Jacobian at each MG level are used to devise efficient Gauss–Seidel smoothers and preferential line iterations are used to eliminate grid anisotropy caused by large aspect ratio elements. The performance of the MG preconditioner is demonstrated in a number of numerical experiments