Methods of Mathematical Physics by Courant, Hilbert PDF

By Courant, Hilbert

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84a) (∇ × E) · ψ dx = 0, ∀ψ ∈ [L (Ω)] . 9). Here also, the magnetic field H is sought in [L 2 (Ω)]3 which is larger than H (curl, Ω) to which belongs H . 10) for instance. , t) ∈ H0 (curl, Ω) and ⎧ 2 d ⎪ ⎪ ⎨ 2 ε E · ϕdx + dt Ω ⎪ ⎪ ⎩ ∀ϕ ∈ H0 (curl, Ω). 2 Functional Issues 19 ⎧ 2 d ⎪ ⎪ ε E · ϕ dx + μ−1 (∇ × E) · (∇ × ϕ) dx ⎪ ⎪ 2 ⎪ dt Ω Ω ⎪ ⎪ ⎪ ⎨ ε d E × n · (ϕ × n) dσ = − jϕ dx, + ⎪ ⎪ ⎪ dt μ ∂Ω Ω ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ ∀ϕ ∈ H (curl, Ω). 82b). 82b) and the boundary integral is ∂Ω [n × (E × n)] · (ψ × n) dσ, ψ ∈ H (curl, Ω).

E. In the following, we assume that the measure of boundary Γ D is not equal to 0. An important consequence is that the norms · 1 and u → ε(u) 0 are equivalent on space H˜ 01 (Ω)3 = {u ∈ H (Ω)3 : u = 0 on Γ D }. This result comes from Korn’s inequality [17]: ∃ CΩ > 0 such that ∀u ∈ H˜ 01 (Ω)3 , u 1 ≤ CΩ ε(u) 0 . 129) This problem is written under the Hille–Yosida theorem framework as follows: 1. H = L 2 (Ω; M) × L 2 (Ω)3 where M is the space of all d × d symmetric matrices. 2 Functional Issues 29 d with u, u˜ 0 = (u i j , u˜ i j )0 .

R −1. The degrees of freedom of the finite element method are the values of the functions of Vhr (R) at these interpolation points. The restriction to an interval [x p , x p+1 ] of a basis function of Vhr (R) corresponding to a degree of freedom located at the point x p, j is defined by the Lagrange polynomial ϕ p, j (x) defined as r +1 x − x p, =1, = j ϕ p, j (x) = , r +1 j = 1 . . r + 1. 1 1D Mass-Lumping and Spectral Elements 41 U T = (u q , u q,1 , . . , u q,r −1 )q∈ZZ . 7c) M1,r is the mass matrix and K 1,r is the stiffness matrix of the discrete problem.

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