Numerical Solution Of The Darwin Model And Applications

The Darwin model is a very good approximation model for the Maxwell's equations when no high frequency or no rapid current change occurs. In 1992, Degond and Raviart in [1] studied the approximation of the Darwin model to the Maxwell's equations in 3-D bounded simply connected domains. They decomposed the electric field E into the sum of E_T and E_L,, where E_T satisfy (?)·E_T = 0, E_L, satisfy (?)×E_L = 0. The Darwin model is obtained from the Maxwell's equations by neglecting(?). They denoted E^D = E_T^D +E_L^D, B^D for the approximations for the field E = E_T + E_L, and B in the Darwin model, they found that || E - E^D ||₀= O(η³), || B - B^D ||₀= O(η²), whereη= (?),Ï…is the characteristic velocity, c is the light velocity.In fact, there are the following result (see [1] ):(1) E_L^D = E_L, = -(?), whereφis the solution of the following problem:(?) (13)whileα_i, 0≤i≤m is the solution of the differential system,(?) (14)Here, x_i satisfy(?) (15)δ_ij=(?) (16)and C_ij stand for (?) depends on the initial condition of E_L,. (2) B^D is the solution of the following problem:(?), (17)(?), (18)(?), (19)(?). (20)(3) E_T^D is the solution of the following problem:(?), (21) (?), (22)(?), (23)(?) (24)In 1997, Ciarlet and Zou [2] studied the solutions of the elliptic boundary value problems in the Darwin model in 3-D bounded simply connected domains. They established the H(curl;Ω) and H(curl. div;Ω) variational formulations, then used Nedelec's and standard finite element methods to solve two kinds of variational problems respectively, they still proved the finite element convergence. In 2003, Ying and Li studied the well-posedness of the electric field in the Darwin model in 2-D unbounded domains in [3], and then solved the problem by the infinite element method, they provided numerical examples as well as convergence analysis. Recently, Fang and Ying studied the Darwin model in 3-D unbounded domains in [19]. They derived the variational formulations, proved the well-posedness and provided some numerical examples.In this paper, Introduction gives the background of the Maxwell's equations and the Darwin model respectively, we introduce the some ideas about the infinite element method, moreover, we describe the detailed process of the infinite element method for the Laplace equation. In the next Chapter 2, we study the Darwin model in 2-D bounded multiply connected domains, we introduce another variable p, establish the mixed variational formulations, we prove the well-posedness and we can prove p=0, and then we use P₂ - P₀ finite element to approximate the variational problem, finally we analyze the convergence. In Chapter 3, we focus on discussing the magnetic field in the Darwin model in 2-D unbounded domains. We firstly establish the variational formulation, prove the well-posedness, and then use the infinite element method to solve the problem, we provide some numerical examples as well as the convergence. Since the solution to the Darwin model is a solution to the Stokes problems with appropriate boundary value and inhomogeneous term, so we recall the infinite element method for the Stokes problem before the magnetic field in the Darwin model. Finally, we describe some results of the electric field in 2-D unbounded domains.In Chapter 4, we study the relationship between the Darwin model and the Maxwell's equations in 2-D bounded multiplied connected domains and the 2-D unbounded domains respectively, it is interesting to find that they are equivalent. It is a big difference from 3-D cases. To that aim, we study the decompositions of the vector fields respectively, the regularity of the solution of the Maxwell's equations, and then we prove the equivalence strictly.In Chapter 5, we consider the relationship between the Maxwell's equations and the Darwin model in 3-D unbounded domains. To that aim, we study a decomposition of the vector fields first of all, and then we neglect (?) in the Maxwell's equations, combining the decomposition of the vector fields, we find that the approximation property between the Maxwell's equations and the Darwin model is the same as in 3-D bounded simply connected domains.In Chapter 6, we consider the adaptive finite element method for Darwin model, and we provide the upper bound estimation based on the posteriori error estimates.