Finite difference approximation of electron balance problem in the stationary high-frequency induction discharges

. The problem of finding the minimal eigenvalue corresponding to a positive eigenfunction of the nonlinear eigenvalue problem for the ordinary differential equation with coefficients depending on a spectral parameter is investigated. This problem arises in modeling the plasma of radio-frequency discharge at reduced pressures. The original differential eigenvalue problem is approximated by the finite difference method on a uniform grid. A sufficient condition for the existence of a minimal eigenvalue corresponding to a positive eigenfunction of the finite difference nonlinear eigenvalue problem is established. Error estimates for the approximate eigenvalue and the corresponding approximate positive eigenfunction are proved. Investigations of this paper generalize well known results for eigenvalue problems with linear dependence on the spectral parameter.


Introduction
In the present paper, we study the nonlinear eigenvalue problem of finding the minimal eigenvalue , We assume that ( ), p µ ( ), r µ , µ ∈ Λ and ( ), s x , x ∈ Ω are smooth positive functions. We also assume that the function ( ), p µ , µ ∈ Λ is nondecreasing and bounded and the function ( ), r µ , µ ∈ Λ is nondecreasing and unbounded. The nonlinear eigenvalue problem (1), (2) is approximated by the finite difference method on a uniform grid. A sufficient condition for the existence of a minimal eigenvalue corresponding to a positive eigenfunction of the finite difference nonlinear eigenvalue problem is established. Error estimates for the approximate eigenvalue and the corresponding approximate positive eigenfunction are proved.
Nonlinear eigenvalue problems of the form (1), (2) arise in modeling the plasma of radio-frequency discharge at reduced pressures. An inductive coupled radio-frequency discharge has found broad applications in diverse technological plasma processes, such as processing textiles and leather-fur half-finished products, metals, hydrogen accumulation by silicon powders, synthesis of oxygen-free ceramic materials, and obtaining carbide and boride materials for nuclear and processing industry [1]. A more effective and qualitative choice of constructive solutions in designing inductive coupled radio-frequency devices requires mathematical models, because some technological characteristics of the plasma cannot be measured.

Variational statement of the problem
For , µ ∈ Λ we introduce the bilinear forms defined by the formulas for any , , u v V ∈ and the Rayleigh ratio ∈ Ω The differential eigenvalue problem (1), (2) is equivalent to the following variational eigenvalue problem: find the least , For fixed , µ ∈ Λ we introduce the following linear parametric variational eigenvalue problem: find the least ( ) , The following variational property is valid be eigenvalues and eigenfunctions satisfying equation (3), ( , ( ), ( )) , The result of Theorem 1 is proved by analogy with [15,16].

Finite difference approximation of the problem
Let us partition the interval [0, ] π by equidistant points , and let h V denote the subspace of the space V consisting of continuous functions h v linear on each element , ∈ Ω We approximate the original eigenvalue problem (10) by a finite-dimensional eigenvalue problem: find the least , For fixed , µ ∈ Λ we introduce the following linear parametric variational eigenvalue problem: find the least ( ) , Then the following variational property is valid then there exists a minimal simple eigenvalue of problem (18) corresponding to a positive eigenfunction.
Proof. Using the variational property of the minimal eigenvalue of problem (19), we derive Then we obtain Then the error estimates of Theorem 3 are proved with using results from [19][20][21][22][23]. This completes the proof of the theorem.
To illustrate the theoretical results of Theorems 1 and 2, we have solved the eigenvalue problem (10)  therefore, the conditions of Theorems 1 and 2 are valid. Fig. 1 shows the graph of the function ( ) γ µ of the parametric eigenvalue problem (11) and the minimal simple eigenvalue 1.3248 λ = of the nonlinear eigenvalue problem (10). We see that the experimental results are consistent with the theoretical results in Theorems 1 and 2. Note that investigations of the present paper generalize well known results for eigenvalue problems with linear dependence on the spectral parameter.