Domain decomposition methods for nonlinear transmission conditions

Nonlinear Overlapping Domain Decomposition Methods

We discuss some overlapping domain decomposition algorithms for solving sparse nonlinear system of equations arising from the discretization of partial differential equations. All algorithms are derived using the three basic algorithms: Newton for local or global nonlinear systems, Krylov for the linear Jacobian system inside Newton, and Schwarz for linear and/or nonlinear preconditioning. The ...

Effective transmission conditions for domain decomposition methods applied to the time-harmonic curl-curl Maxwell's equations

The time-harmonic Maxwell equations describe the propagation of electromagnetic waves and are therefore fundamental for the simulation of many modern devices we have become used to in everyday life. The numerical solution of these equations is hampered by two fundamental problems: first, in the high frequency regime, very fine meshes need to be used in order to avoid the pollution effect well k...

Optimized interface conditions for domain decomposition methods in fluid dynamics

Interface conditions (IC) between subdomains have an important impact on the convergence rate of domain decomposition algorithms. We rst recall the Schwarz method which is based on the use of Dirichlet conditions on the boundaries of the subdomains and overlapping subdomains. We explain how it is possible to replace them by more e cient ICs with normal and tangential derivatives so that overlap...

Optimization Based Domain Decomposition Methods for Linear and Nonlinear Problems

(ABSTRACT) Optimization based domain decomposition methods for the solution of partial differential equations are considered. The crux of the method is a constrained minimization problem for which the objective functional measures the jump in the dependent variables across the common boundaries between subdomains; the constraints are the partial differential equations. First, we consider a line...

Domain Decomposition Methods with Coupled Transmission Conditions for the Optimal Control of Systems Governed by Elliptic Partial Differential Equations