This paper proposes a multiscale method for solving the numerical solution of mean field games which accelerates convergence and addresses problem determining initial guess. Starting from an approximate at coarsest level, constructs approximations on successively finer grids via alternating sweeping, not only allows use classical time marching schemes, but also enables applications to both loca...

We present an energy-preserving mechanic formulation for dynamic quasi-brittle fracture in Eulerian–Lagrangian formulation, where a second-order phase-field equation controls the damage evolution. The numerical adapts space and time to bound errors, solving mesh-bias issues these models typically suffer. time-step adaptivity estimates temporal truncation error of partial differential that gover...