Optimized Schwarz waveform relaxation and mixed hybrid finite element methods for transport problems in porous media

Abstract: This talk is concerned with numerical schemes for linear advection-diffusion problems, in which different time steps (with no CFL restrictions) can be used in different parts of the domain. A mixed formulation is considered where the flux variable represents the total flux (i.e., both diffusive and advective flux, instead of only diffusive flux as in standard mixed methods). Based on nonoverlapping domain decomposition and optimized Schwarz waveform relaxation (OSWR), the problem on the whole domain is reformulated as local problems in the subdomains with optimized Robin or Ventcel transmission conditions on the interfaces between the subdomains. Using a substructuring technique, the coupled subdomain problems are reduced to a space-time interface problem which is solved iteratively; each iteration involves solving the subdomain problems (with Robin or Ventcel boundary conditions) independently and globally in time. The subdomain problems are discretized in space by a mixed hybrid finite element method based on the lowest-order Raviart-Thomas space and in time by the backward Euler method. The proposed algorithms are fully implicit and enable local time discretizations in the subdomains. Convergence analysis of the fully discrete OSWR method with Robin conditions is presented. Numerical results with discontinuous coefficients and various Peclét numbers illustrate the performance of the methods on nonconforming time grids and compare the convergence of optimized Robin and Ventcel transmission conditions.

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