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

Optimized Ventcel-Schwarz waveform relaxation and mixed hybrid finite element method for transport problems

“…Theoretical results on how to choose such a ratio for different coefficients and error analysis of nonconforming time grids have not yet been established. In practice, the ratio between the coarse and fine time steps is determined on a case-by-case basis -depending on the jumps in the coefficients as shown numerically in [31,34,30] and the next section. We also remark that global-in-time domain decomposition methods are implicit in time, thus large time steps can be used (without affecting the numerical stability) when the model parameters are small [35].…”

“…The OSWR methods have been extensively studied mostly for the primal formulation with either Lagrange finite element or finite volume discretizations. In the context of mixed formulations, OSWR methods with Robin transmission conditions and nonconforming time grids were studied for pure diffusion problems in [31] and for the advection-diffusion problems in [34,30]. Operator splitting was used in [34] so that the advection is treated explicitly and the diffusion implicitly, while the method in [30] is fully implicit in time and the problem is discretized in space by mixed hybrid finite elements [40,10].…”

“…In the context of mixed formulations, OSWR methods with Robin transmission conditions and nonconforming time grids were studied for pure diffusion problems in [31] and for the advection-diffusion problems in [34,30]. Operator splitting was used in [34] so that the advection is treated explicitly and the diffusion implicitly, while the method in [30] is fully implicit in time and the problem is discretized in space by mixed hybrid finite elements [40,10]. In [33], OSWR methods were applied to a reduced fracture model of the flow of a compressible fluid in a porous medium in which the fracture is treated as an interface between two subdomains and the so-called Ventcel-to-Robin transmission conditions were derived for such a model.…”

“…The parameters α i,j and β i,j are chosen to optimize the convergence factor as studied in [5,27]. Note that the transmission conditions (6) reduce to Robin transmission conditions [30] when β i,j = 0.…”

“…Note that for the space-discrete advection term in (19), we have used the Lagrange multiplier instead of using the piecewise constant concentration. Such a scheme is shown to give good numerical performance for the case where advection is moderately dominant [40,30]. For strongly advection-dominated problems, using upwind values are recommended; interested readers are referred to [40,10] for further details.…”