Thats a unique idea.You should consider the following idea to enhance your idea so that it would impact uniquely: The BDDC algorithm is extended to a large class of discontinuous Galerkin (DG) discretizations of second order elliptic problems. An estimate of $C(1+\log(H/h))^2$ is obtained for the condition number of the preconditioned system where $C$ is a constant independent of $h$ or $H$ or large jumps in the coefficient of the problem. Numerical simulations are presented which confirm the theoretical results. A key component for the development and analysis of the BDDC algorithm is a novel perspective presenting the DG discretization as the sum of elementwise „local” bilinear forms. The elementwise perspective allows for a simple unified analysis of a variety of DG methods and leads naturally to the appropriate choice for the subdomainwise local bilinear forms. Additionally, this new perspective enables a connection to be drawn beten the DG discretization and a related continuous finite element discretization to simplify the analysis of the BDDC algorithm.
['Laslo Tibor Diosady', 'David L. Darmofal']
I am writing a research paper on a idea A Unified Analysis of Balancing Domain Decomposition by Constraints for Discontinuous Galerkin Discretizations
[{"from": "human", "value": "I am writing a research paper on a idea A Unified Analysis of Balancing Domain Decomposition by Constraints for Discontinuous Galerkin Discretizations"}, {"from": "gpt", "value": "Thats a unique idea.You should consider the following idea to enhance your idea so that it would impact uniquely: The BDDC algorithm is extended to a large class of discontinuous Galerkin (DG) discretizations of second order elliptic problems. An estimate of $C(1+\\log(H/h))^2$ is obtained for the condition number of the preconditioned system where $C$ is a constant independent of $h$ or $H$ or large jumps in the coefficient of the problem. Numerical simulations are presented which confirm the theoretical results. A key component for the development and analysis of the BDDC algorithm is a novel perspective presenting the DG discretization as the sum of elementwise \u201elocal\u201d bilinear forms. The elementwise perspective allows for a simple unified analysis of a variety of DG methods and leads naturally to the appropriate choice for the subdomainwise local bilinear forms. Additionally, this new perspective enables a connection to be drawn beten the DG discretization and a related continuous finite element discretization to simplify the analysis of the BDDC algorithm."}]