To deal with the divergence-free constraint in a double curl problem: curl µ −1 curl u = f and div εu = 0 in Ω, where µ and ε represent the physical properties of the materials occupying Ω, we develop a δ-regularization method: curl µ −1 curl u δ + δεu δ = f to completely ignore the divergence-free constraint div εu = 0. It is shown that u δ converges to u in H(curl ; Ω) norm as δ → 0. The edge finite element method is then analyzed for solving u δ. With the finite element solution u δ,h , quasi-optimal error bound in H(curl ; Ω) norm is obtained between u and u δ,h , including a uniform (with respect to δ) stability of u δ,h in H(curl ; Ω) norm. All the theoretical analysis is done in a general setting, where µ and ε may be discontinuous, anisotropic and inhomogeneous, and the solution may have a very low piecewise regularity on each material subdomain Ω j with u, curl u ∈ (H r (Ω j)) 3 for some 0 < r < 1, where r may be not greater than 1/2. To establish the uniform stability and the error bound for r ≤ 1/2, we have respectively developed a new theory for the K h ellipticity (related to mixed methods) and a new theory for the Fortin interpolation operator. Numerical results presented confirm the theory.
In view of the problem that the predictive results of flow quantity are not ideal for the predictive models at present. Based on the chaos identification to the flood system, chaos BP neural network model are developed combined chaos theory and BP neural netwok, flood sequences are disposed by phase-space reconstruction to be as training sample. Network structure can be determined by Matlab toolbox. The established chaos BP model is used to predict the phenomenon of peak value for Huayuankou hydrometric station in 2006. The results show that the predictive model combined chaos theory and BP neural network, has certain reference value to improve flood forecasting accuracy as a new attempt.
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