On the Schwarz alternating method with more than two subdomains for nonlinear monotone problems
SIAM Journal on Numerical Analysis
Monotone multigrid methods for elliptic variational inequalities I
Numerische Mathematik
Domain decomposition: parallel multilevel methods for elliptic partial differential equations
Domain decomposition: parallel multilevel methods for elliptic partial differential equations
Iterative algorithms of domain decomposition for the solution of a quasilinear elliptic problem
Journal of Computational and Applied Mathematics
Rate of Convergence of Some Space Decomposition Methods for Linear and Nonlinear Problems
SIAM Journal on Numerical Analysis
Schwarz algorithm for the solution of variational inequalities with nonlinear source terms
Applied Mathematics and Computation
On Schwarz Alternating Methods for Nonlinear Elliptic PDEs
SIAM Journal on Scientific Computing
Finite Element Method for Elliptic Problems
Finite Element Method for Elliptic Problems
On Schwarz Alternating Methods for the Incompressible Navier--Stokes Equations
SIAM Journal on Scientific Computing
Global and uniform convergence of subspace correction methods for some convex optimization problems
Mathematics of Computation
Convergence Rate Analysis of a Multiplicative Schwarz Method for Variational Inequalities
SIAM Journal on Numerical Analysis
SIAM Journal on Numerical Analysis
| Hi-index | 7.29 |
We prove the convergence of some multiplicative and additive Schwarz methods for inequalities which contain contraction operators. The problem is stated in a reflexive Banach space and it generalizes the well-known fixed-point problem in the Hilbert spaces. Error estimation theorems are given for three multiplicative algorithms and two additive algorithms. We show that these algorithms are in fact Schwarz methods if the subspaces are associated with a decomposition of the domain. Also, for the one- and two-level methods in the finite element spaces, we write the convergence rates as functions of the overlapping and mesh parameters. They are similar with the convergence rates of these methods for linear problems. Besides the direct use of the five algorithms for the inequalities with contraction operators, we can use the above results to obtain the convergence rate of the Schwarz method for other types of inequalities or nonlinear equations. In this way, we prove the convergence and estimate the error of the one- and two-level Schwarz methods for some inequalities in Hilbert spaces which are not of the variational type, and also, for the Navier-Stokes problem. Finally, we give conditions of existence and uniqueness of the solution for all problems we consider. We point out that these conditions and the convergence conditions of the proposed algorithms are of the same type.