The Maximum Norm Analysis of Schwarz Method for Elliptic Quasi-Variational Inequalities

In this paper, we present a maximum norm analysis of an overlapping Schwartz method on non matching grids for a quasi-variational inequality, where the obstacle and the second member depend on the solution. Our result improves and generalizes some previous results.

Optimized Schwarz methods with general Ventcell transmission conditions for fully anisotropic diffusion with discrete duality finite volume discretizations

We introduce a new non-overlapping optimized Schwarz method for fully anisotropic diffusion problems. Optimized Schwarz methods take into account the underlying physical properties of the problem at hand in the transmission conditions, and are thus ideally suited for solving anisotropic diffusion problems. We first study the new method at the continuous level for two subdomains, prove its convergence for general transmission conditions of Ventcell type using energy estimates, and also derive convergence factors to determine the optimal choice of parameters in the transmission conditions. We then derive optimized Robin and Ventcell parameters at the continuous level for fully anisotropic diffusion, both for the case of unbounded and bounded domains. We next present a discretization of the algorithm using discrete duality finite volumes, which are ideally suited for fully anisotropic diffusion on very general meshes. We prove a new convergence result for the discretized optimized Schwarz method with two subdomains using energy estimates for general Ventcell transmission conditions. We finally study the convergence of the new optimized Schwarz method numerically using parameters obtained from the continuous analysis. We find that the predicted optimized parameters work very well in practice, and that for certain anisotropies which we characterize, our new bounded domain analysis is important.

A Schwarz iterative method to evaluate ocean–atmosphere coupling schemes: implementation and diagnostics in IPSL-CM6-SW-VLR

State-of-the-art Earth system models, like the ones used in the Coupled Model Intercomparison Project Phase 6 (CMIP6), suffer from temporal inconsistencies at the ocean–atmosphere interface. Indeed, the coupling algorithms generally implemented in those models do not allow for a correct phasing between the ocean and the atmosphere and hence between their diurnal cycles. A possibility to remove these temporal inconsistencies is to use an iterative coupling algorithm based on the Schwarz iterative method. Despite its large computational cost compared to standard coupling methods, which makes the algorithm implementation impractical for production runs, the Schwarz method is useful to evaluate some of the errors made in state-of-the-art ocean–atmosphere coupled models (e.g., in the representation of the processes related to diurnal cycle), as illustrated by the present study. IPSL-CM6-SW-VLR is a low-resolution version of the IPSL-CM6 coupled model with a simplified land surface model, implementing a Schwarz iterative coupling scheme. Comparisons between coupled solutions obtained with this new scheme and the standard IPSL coupling scheme (referred to as the parallel algorithm) show large differences after sunrise and before sunset, when the external forcing (insolation at the top of the atmosphere) has the fastest pace of change. At these times of the day, the difference between the two numerical solutions is often larger than 100 % of the solution, even with a small coupling period, thus suggesting that significant errors are potentially made with current coupling methods. Most of those differences can be strongly reduced by making only two iterations of the Schwarz method, which leads to a doubling of the computing cost. Besides the parallel algorithm used in IPSL-CM6, we also test a so-called sequential atmosphere-first algorithm, which is used in some coupled ocean–atmosphere models. We show that the sequential algorithm improves the numerical results compared to the parallel one at the expanse of a loss of parallelism. The present study focuses on the ocean–atmosphere interface with no sea ice. The problem with three components (ocean–sea ice–atmosphere) remains to be investigated.

Non-local variant of the Optimised Schwarz Method for arbitrary non-overlapping subdomain partitions

We consider a scalar wave propagation in harmonic regime modelled by Helmholtz equation with heterogeneous coefficients. Using the Multi-Trace Formalism (MTF), we propose a new variant of the Optimized Schwarz Method (OSM) that remains valid in the presence of cross-points in the subdomain partition. This leads to the derivation of a strongly coercive formulation of our Helmholtz problem posed on the union of all interfaces. The corresponding operator takes the form "identity + non-expansive".