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This paper is concerned with numerical solution of transport problems inheterogeneous porous media. A semi-discrete continuous-in-time formulation ofthe linear advection-diffusion equation is obtained by using a mixed hybridfinite element method, in which the flux variable represents both the advectiveand diffusive flux, and the Lagrange multiplier arising from the hybridizationis used for the discretization of the advective term. Based on global-in-timeand nonoverlapping domain decomposition, we propose two implicit localtime-stepping methods to solve the semi-discrete problem. The first method usesthe time-dependent Steklov-Poincar\'e type operator and the second uses theoptimized Schwarz waveform relaxation (OSWR) with Robin transmissionconditions. For each method, we formulate a space-time interface problem whichis solved iteratively. Each iteration involves solving the subdomain problemsindependently and globally in time; thus, different time steps can be used inthe subdomains. The convergence of the fully discrete OSWR algorithm withnonmatching time grids is proved. Numerical results for problems with variousPecl\'et numbers and discontinuous coefficients, including a prototype for thesimulation of the underground storage of nuclear waste, are presented toillustrate the performance of the proposed local time-stepping methods.

Fully implicit local time-stepping methods for advection-diffusion problems in mixed formulations