This paper is concerned with numerical solution of transport problems in heterogeneous porous media. A semi-discrete continuous-in-time formulation of the linear advection-diffusion equation is obtained by using a mixed hybrid finite element method, in which the flux variable represents both the advective and diffusive flux, and the Lagrange multiplier arising from the hybridization is used for the discretization of the advective term. Based on global-in-time and nonoverlapping domain decomposition, we propose two implicit local time-stepping methods to solve the semi-discrete problem. The first method uses the time-dependent Steklov-Poincaré type operator and the second uses the optimized Schwarz waveform relaxation (OSWR) with Robin transmission conditions. For each method, we formulate a space-time interface problem which is solved iteratively. Each iteration involves solving the subdomain problems independently and globally in time; thus, different time steps can be used in the subdomains. The convergence of the fully discrete OSWR algorithm with nonmatching time grids is proved. Numerical results for problems with various Peclét numbers and discontinuous coefficients, including a prototype for the simulation of the underground storage of nuclear waste, are presented to illustrate the performance of the proposed local time-stepping methods.

本文关注非均质多孔介质中输运问题的数值解。采用混合混合有限元方法得到线性平流-扩散方程的半离散时间连续公式，其中通量变量代表平流通量和扩散通量，由杂交产生的拉格朗日乘数为用于平流项的离散化。基于全局时间和非重叠域分解，我们提出了两种隐式局部时间步长方法来解决半离散问题。第一种方法使用时间相关的 Steklov-Poincaré 类型算子，第二种方法使用具有 Robin 传输条件的优化 Schwarz 波形弛豫 (OSWR)。对于每种方法，我们制定了一个迭代求解的时空界面问题。每次迭代都涉及在时间上独立且全局地解决子域问题；因此，可以在子域中使用不同的时间步长。证明了时间网格不匹配的全离散OSWR算法的收敛性。给出了具有各种 Peclét 数和不连续系数的问题的数值结果，包括用于模拟地下核废料储存的原型，以说明所提出的局部时间步长方法的性能。