Journal of Computational Physics
Optimized Domain Decomposition Methods for the Spherical Laplacian
SIAM Journal on Numerical Analysis
The Optimized Schwarz Method with a Coarse Grid Correction
SIAM Journal on Scientific Computing
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A nonoverlapping domain decomposition iterative procedure is developed and analyzed for second order elliptic problems in $\mathbb R^N$. Its convergence is proved. The method is based on a Robin-type consistency condition with two parameters, called a transmission coefficient and a penalty coefficient, as a transmission condition together with a derivative-free transmission data updating technique on the artificial interfaces. Then the method is applied to the nonconforming finite element problems. A nonoverlapping domain decomposition iterative procedure for solving the nonconforming finite element problems of second order partial differential equations is developed and analyzed, which is directly presented to the nonconforming finite element problems without introducing any Lagrange multipliers. Its convergence is demonstrated, and the convergence rate is derived. The convergence analyses imply that the convergence rate is independent of the finite element meshes size while choosing the right parameters. Furthermore, the conclusions are extended to the unstructured finite element meshes. For both continuous problems and discrete problems, the method of this paper can be applied to general multisubdomain decompositions and implemented on parallel machines with local communications naturally. The method also allows choosing subdomains very flexibly, even as small as an individual element for finite element problems.