Title:

Analysis and massively parallel implementation of the 2Lagrange multiplier methods and optimized Schwarz methods

A partial diﬀerential equation (PDE) is an equation that involves partial derivatives of an unknown function u : Ω →R. The domain Ω denotes an open subset of Rd, d = 2,3. Partial diﬀerential equations (PDEs) are used in order to model problems in all areas of science, including physics, engineering, ﬁnance, etc... It is often impossible to solve a PDE exactly using analytical methods and thus most PDEs are solved numerically. In order to solve a PDE numerically, one introduces a triangulation of the domain with ﬁnitely many vertices. The solution of the PDE is approximated by a piecewise polynomial function on this grid; this is the Finite Element Method (FEM). For linear PDEs, the FEM leads to a linear system of equations of the form Au = f, where A is a very large n×n matrix that is very sparse. Solving Au = f using Gaussian elimination results in signiﬁcant “ﬁllin”, where many zero entries of A become nonzero as the Gaussian elimination algorithm progresses. For very large problems, Gaussian elimination will exhaust the main memory of any computer. As a result, it is necessary to use an iterative algorithm to solve the problem Au = f. Classical iterative methods such as Jacobi and GaussSeidel require more and more iterations as the size n of A increases and thus these iterations are not useful for large values of n. Domain decomposition is a more sophisticated iterative scheme based on partitioning the domain Ω into many subdomains {Ωk}. Since the Green’s functions are nonlocal, it is impossible to solve local problems on the subdomains Ωk and thus obtain a global solution unless we iterate by exchanging information between the local problems.
In the method of H. Schwarz, the information exchanged across the “artiﬁcial interfaces” is Dirichlet data. J. L. Lions improved the Schwarz method by exchanging Robin boundary data. Gander and his collaborators found that one could obtain accelerated convergence by tuning the Robin parameter; this is the optimized Schwarz method (OSM). The design and analysis of optimized Schwarz methods for general domains and subdomains has proven to be a challenge. Loisel found that the OSM is dual to the 2Lagrange multiplier (2LM) method, which is an iteration on the Robin traces; the 2Lagrange multiplier method is amenable to analysis. OSM with a coarse grid correction has previously been considered for cylindrical domains using Fourier analysis. This approach has some serious limitations, since only very regular domains and subdomains and only the Laplacian can be considered. These limitations are not purely theoretical; the implementation of an OSM for a domain decomposition with cross points (points that are shared between three or more subdomains) has remained challenging. Our main result is the design and implementation of a 2Lagrange multiplier method, which is dual to the OSM, including a coarsegrid correction, which handles cross points. This is highly valuable not only from the point of view of analysis, but also because it describes in detail how the cross points are handled by the implementation. Our analysis shows that the condition number of the linear problem scales like O((H/h)1/2), where H is the subdomain diameter and h is the grid parameter. We have implemented our algorithms in C using the PETSc library. Numerical experiments performed on the HECToR supercomputer conﬁrm the good scaling properties of our algorithms.
