Thats a unique idea.You should consider the following idea to enhance your idea so that it would impact uniquely:  propose a discrete ighted Helmholtz decomposition for edge element functions. The decomposition is orthogonal in a ighted $$L^2$$ inner product and stable uniformly with respect to the jumps in the discontinuous ight function. As an application, the new Helmholtz decomposition is applied to demonstrate the quasi-optimality of a preconditioned edge element system for solving a saddle-point Maxll system in non-homogeneous media by a non-overlapping domain decomposition preconditioner, i.e., the condition number grows only as the logarithm of the dimension of the local subproblem associated with an individual subdomain, and more importantly, it is independent of the jumps of the physical coefficients across the interfaces beten any two subdomains of different media. Numerical experiments are presented to validate the effectiveness of the non-overlapping domain decomposition preconditioner.
['Qiya Hu', 'Shi Shu', 'Jun Zou']
I am writing a research paper on a idea A discrete weighted Helmholtz decomposition and its application
[{"from": "human", "value": "I am writing a research paper on a idea A discrete weighted Helmholtz decomposition and its application"}, {"from": "gpt", "value": "Thats a unique idea.You should consider the following idea to enhance your idea so that it would impact uniquely:  propose a discrete ighted Helmholtz decomposition for edge element functions. The decomposition is orthogonal in a ighted $$L^2$$ inner product and stable uniformly with respect to the jumps in the discontinuous ight function. As an application, the new Helmholtz decomposition is applied to demonstrate the quasi-optimality of a preconditioned edge element system for solving a saddle-point Maxll system in non-homogeneous media by a non-overlapping domain decomposition preconditioner, i.e., the condition number grows only as the logarithm of the dimension of the local subproblem associated with an individual subdomain, and more importantly, it is independent of the jumps of the physical coefficients across the interfaces beten any two subdomains of different media. Numerical experiments are presented to validate the effectiveness of the non-overlapping domain decomposition preconditioner."}]