Topic
Neumann–Dirichlet method
About: Neumann–Dirichlet method is a research topic. Over the lifetime, 380 publications have been published within this topic receiving 6241 citations.
Papers published on a yearly basis
1 / 2
Papers
TL;DR: Recent developments in the non-standard asymptotics of the narrow escape problem are reviewed, which are based on several ingredients: a better resolution of the singularity of Neumann's function,resolution of the boundary layer near the small target by conformal mappings of domains with bottlenecks, and the breakup of composite domains into simpler components.
Abstract: The narrow escape problem in diffusion theory is to calculate the mean first passage time of a diffusion process to a small target on the reflecting boundary of a bounded domain. The problem is equivalent to solving the mixed Dirichlet--Neumann boundary value problem for the Poisson equation with small Dirichlet and large Neumann parts. The mixed boundary value problem, which goes back to Lord Rayleigh, originates in the theory of sound and is closely connected to the eigenvalue problem for the mixed problem and for the Neumann problem in domains with bottlenecks. We review here recent developments in the non-standard asymptotics of the problem, which are based on several ingredients: a better resolution of the singularity of Neumann's function, resolution of the boundary layer near the small target by conformal mappings of domains with bottlenecks, and the breakup of composite domains into simpler components. The new methodology applies to two- and higher-dimensional problems. Selected applications are r...
200 citations
192 citations
TL;DR: A substructuring iterative method with Lagrange multipliers decomposes finite element discretization of an elliptic boundary value problem into Neumann problems on the subdomains plus a coarse problem for the subdomain nullspace components and proves the asymptotic bound on the condition number 1.
Abstract: We analyze the convergence of a substructing iterative method with Lagrange multipliers, propose recently by Farhat and Roux. The method decomposes finite element discretization of an elliptic boundary value problem into Neumann problems on the subdomains plus a coarse problem for the subdomain nullspace components. For linear conforming elements and preconditioning by the Dirichlet problems on the subdomains, we prove the asymptotic bound on the condition number C(1+log(H/h))^\alpha, \alpha=2 or 3, where h is the characteristic element size, H subdomain size, and C is independent of the number of subdomains.
180 citations
Book•
12 Dec 1990
TL;DR: In this paper, a generalization of Dirichlet branches bifurcating from zero to zero is presented. And the properties of time maps and period maps are discussed. But they are not discussed in detail.
Abstract: Dirichlet branches bifurcating from zero.- Neumann problems, period maps and semilinear dirichlet problems.- Generalizations.- General properties of time maps.
156 citations
TL;DR: The FETI-DPEM method is applied to the electromagnetic simulation of array-type structures where the geometrical redundancy is fully exploited to speedup the simulation and reduce the memory requirement.
Abstract: A Lagrange multiplier based non-overlapping domain decomposition method, referred to as the dual-primal finite element tearing and interconnecting (FETI-DP), is formulated for the finite element simulation of large, three-dimensional (3-D) electromagnetic problems. This formulation extends the FETI-DP for solving the scalar Helmholtz equation to the solution of the vector curl-curl wave equation using edge-based finite elements. It enforces the field continuity explicitly along the edges shared by more than two subdomains and implicitly at the interfaces between two subdomains through the use of Lagrange multipliers. With the aid of a direct sparse solver for each subdomain system, the large global problem is reduced to a much smaller interface problem, from which a Neumann boundary condition is obtained at the interfaces between all the subdomains. This Neumann boundary condition is then used to calculate the field within each subdomain. It is shown that the resulting FETI-DPEM method is scalable with respect to the size of finite elements and the number of subdomains. It is also scalable with respect to the size of the subdomains when the subdomains, with its surfaces enclosed by perfect magnetic conductors, cannot support any resonant modes. The FETI-DPEM method is applied to the electromagnetic simulation of array-type structures where the geometrical redundancy is fully exploited to speedup the simulation and reduce the memory requirement. Numerical results for the simulation of finite antenna arrays and photonic bandgap devices are presented to demonstrate the application, accuracy, efficiency, and capability of the FETI-DPEM method
155 citations
Related Topics (5)
Performance Metrics
| Year | Papers |
|---|---|
| 2021 | 1 |
| 2018 | 1 |
| 2017 | 14 |
| 2016 | 11 |
| 2015 | 9 |
| 2014 | 17 |