WELL-POSEDNESS OF THE DIFFERENCE SCHEMES OF THE HIGH ORDER OF ACCURACY FOR ELLIPTIC EQUATIONS

It is well known the differential equation − u ″ ( t ) + A u ( t ) = f ( t ) ( − ∞ t ∞ ) in a general Banach space E with the positive operator A is ill-posed in the Banach space C ( E ) = C ( ( − ∞ , ∞ ) , E ) of the bounded continuous functions ϕ ( t ) defined on the whole real line with norm ‖ ϕ ‖ C ( E ) = sup ⁡ − ∞ t ∞ ‖ ϕ ( t ) ‖ E . In the present paper we consider the high order of accuracy two-step difference schemes generated by an exact difference scheme or by Taylor's decomposition on three points for the approximate solutions of this differential equation. The well-posedness of these difference schemes in the difference analogy of the smooth functions is obtained. The exact almost coercive inequality for solutions in C ( τ , E ) of these difference schemes is established.


Introduction
The role played by coercive inequalities in the study of local boundary-value problems for elliptic and parabolic differential equations is well known (see, e.g., [10,11,16]).
Coercivity inequalities in Hölder norms with a weight for the solutions of an abstract differential equation of elliptic type were established for the first time by Sobolevskiȋ [12].Further in [1,2,4,7,13] the coercive inequalities in Hölder norms with a weight were obtained for the solutions of various local and nonlocal boundary value problems for differential and difference equations of elliptic type.Finally in [3,5,8,14] the corresponding coercivity inequalities in Hölder norms without a weight for the solutions of some local and nonlocal boundary value problems were obtained.
In the present paper we consider the differential equation in an arbitrary Banach space E with positive operator A. (i) u(t) is the bounded twice continuously differentiable on the real line (−∞,∞); (ii) the element u(t) belongs to D(A) for all t ∈ (−∞,∞), and the function Au(t) is bounded continuous for all t ∈ (−∞,∞); (iii) u(t) satisfies (1.1).A solution of the differential equation (1.1) defined in this manner will from now on referred to as a solution of the differential equation (1.1) in the space C(E) = C((−∞,∞), E) of all bounded continuous functions ϕ(t) defined on (−∞,∞) with values in E equipped with the norm (1. 2) The well-posedness in C(E) of the differential equation (1.1) means that coercive inequality is true for its solution u(t) ∈ C(E) with some M, does not depend on f (t) ∈ C(E).It is known that from the coercive inequality (1.3) it follows the positivity of the operator A in the Banach space E, that means the operator Iλ + A has bounded in E inverse (Iλ + A) −1 for any λ ≥ 0, and estimate holds for some 1 ≤ M < ∞.It turns out that this positivity property of the operator A in E is necessary condition of well-posedness of the differential equation (1.1) in C(E).The positivity of the operator A in E is not a sufficient condition for the well-posedness of the differential equation (1.1).As it turns out, problem (1.1) is not well-posed for all such operators.The counterexample given by Sobolevskiȋ [14].
It is known (see, e.g, [9,12]) that the operator A 1/2 has better spectral properties than the positive operator A. In particular, the operator λI + √ A has a bounded inverse for any complex number λ with Reλ ≥ 0, and the estimate λI is true for some M ≥ 1.Thus, √ A is a strongly positive in E operator, that is, the following estimates hold: That means if spectral angle ϕ(A,E) of operator A in E equal π, then spectral angle ϕ(A (Here and in this paper we will put B = A 1/2 ) gives a solution of differential equation (1.1) in C(E).
In the papers [8,14] the well-posedness of the differential equation (1.1) was established in the Banach space C α (E) = C α ((−∞,∞),E) (0 < α < 1) obtained by completion of the set of E-valued smooth functions ϕ(t) on (−∞,∞) in the norm and in the Banach space Here, the Banach space E α = E α (E,A 1/2 ) (0 < α < 1) consists of those v ∈ E for which the norm (see, e.g, [6]) is finite.Moreover, in the paper [14] the difference scheme second order of accuracy for the approximate solutions of (1.1) was studied.The exact almost coercive inequality for solutions in C(τ,E) of this difference scheme was established.
In the present paper we consider two-step difference schemes of the high order of accuracy generated by an exact difference scheme or by Taylor's decomposition on three points for the approximate solutions of (1.1).The well-posedness of these difference schemes in the difference analogy of the smooth functions is obtained.The exact almost coercive inequality for solutions in C(τ,E) of these difference schemes is established.

Difference schemes generated by an exact difference scheme
The construction of two-step difference schemes of an arbitrary high order of accuracy for the approximate solutions of the differential equation (1.1) is based on the following theorem.

Theorem 2.1. Let u(t k ) be a solution of the differential equation (1.1) at the grid points
is the solution of the following second order difference equations: 4 Well-posedness of the difference schemes The proof of Theorem 2.1 is based on the formula (1.7) for the solution of the differential equation (1.1).
From (2.1) it is clear that for approximate solutions of the differential equation (1.1) it is necessary to approximate the expressions exp{−τB} and Let us remark that in constructing difference schemes it is important to know how to construct a right-hand side ψ l, j k that satisfies and is sufficiently simple.The choice formula ( Applying the exact difference scheme (2.1) and formulas (2.4), we obtain (l + j)-order of accuracy two-step difference schemes (2.5) Here, the function R j,l (z) is constructed on the base of Padé's fractions respectively where the coefficients a i , a i = 1,..., j, and b i , i = 1,...,l, are uniquely defined from the A. Ashyralyev and P. E. Sobolevskiȋ 5 condition for |z| → 0.
Let F(τ,E) be the linear space of mesh functions ϕ τ = {ϕk} ∞ −∞ with values in the Banach space E. Next on F(τ,E) we denote C(τ,E) and C α (τ,E) Banach spaces with the norms (2.9) We say that the difference scheme (2.5) is stable in F(τ,E) if we have the inequality where M is independent not only of ψ τ j,l , but also on τ.We say that the difference scheme (2.5) is coercively stable (well-posed) in F(τ,E) if we have the coercive inequality where M is independent not only of ψ τ j,l but also of τ.It is known that if A is a positive operator, then A 1/2 is a strongly positive operator.Since, difference schemes (2.5) are constructed by using an operator A 1/2 , instead of A. Therefore, we have the following theorem.
Theorem 2.2.Let j = l − 1,l − 2. Then two-step difference schemes (2.5) Since the differential equation (1.1) in the space C(E) of bounded continuous functions defined on the real line with values in E is not well-posed in the case of general positive operator A, then the well-posedness of the difference schemes (2.5) in C(τ,E) norm does not take place uniformly with respect to τ > 0. This means that the coercive norm tends to ∞ as τ → +0.The investigation of difference schemes (2.5) permits us to establish the order of growth of this norm to ∞.

Difference schemes generated by the Taylor's decomposition
Let We consider two-step difference schemes of 2l-order of accuracy of approximately solving (1.1).Difference schemes (3.1) generated by an exact difference scheme This exact difference scheme generated by the Taylor's decomposition on three points.This difference equation is uniquely solvable, and the following formula holds where Actually, the difference scheme (3.1) can be rewritten as the equivalent system of the first order linear difference equations with operator B = B(τ,A).Note that B(τ,A) = A 1/2 but then B(τ,A) → A 1/2 as τ → 0 and it has same spectral properties of A 1/2 under the strong positivity assumption for A. From the last system it follows the system of recursion formulas where R(τB) = (I + τB) −1 .Hence From that it follows that we have that From this formula it follows the formula (3.3).
As above, let us give the definition of the stability and coercively stable of the difference equation (3.1).
The difference equation (3.1) is said to be stable in F τ (E) if we have the inequality where M is independent not only of ϕ τ but also of τ.
The difference equation (3.1) is said to be well-posed (coercively stable) in F τ (E) if we have the coercive inequality where M is independent not only of ϕ τ but also of τ.
From the formula (3.3) it follows that the investigation of the stability and wellposedness of difference schemes (3.1) relies in an essential manner on a number of properties of the powers of the operator (I + τB) −1 .We are not able to obtain the estimates for powers of the operator (I + τB) −1 in the general cases of operator A. However, for the investigation of (3.1) it is necessary to construct an operator B = B(τ,A) and to give estimates The proof of estimates (3.13) is based upon three theorems.
Theorem 3.1 (a real criterion for strong positivity [13]).A necessary and sufficient condition for B to be strongly positive is that the estimates (3.13) are satisfied.

Theorem 3.2 ([13]
).If A is a strongly positive operator, then the operator B denoted by is a strongly positive operator.
).If A is a strongly positive operator with spectrum angle φ(A,E) ≤ π/2l then the operator A l denoted by formula (3.1) is also a strongly positive operator.
We have the following results.
Theorem 3.4.Let A is a strongly positive operator in a Banach space E with spectral angle φ(A,E) < π/2l.Then difference problem (3.1) is stable in C α (τ,E), (0 ≤ α ≤ 1).For the solutions of the difference problem (3.1) satisfy the stability inequalities where M does not depend on ϕ τ , α and τ.
A. Ashyralyev and P. E. Sobolevskiȋ 9 Theorem 3.5.Let A is a strongly positive operator in a Banach space E with spectral angle φ(A,E) < π/2l.Then the solutions of the difference problem (3.1) in C(τ,E) obey the almost coercive inequality where M 1 is independent not only of ϕ τ , but also of τ.
Theorem 3.6.Let A is a strongly positive operator in a Banach space E with spectral angle φ(A,E) < π/2l.Then the solutions of the difference problem (3.1) in C α (τ,E) and C(τ,E α ), 0 < α < 1 obey the coercive inequality where M 1 is independent not only of ϕ τ , α, but also of τ.
It is clear that From this estimate and Theorems 3.5-3.6 it follows that the following theorem.
Theorem 3.7.Let A is a strongly positive operator in a Banach space E with spectral angle φ(A,E) < π/2l.Then the solutions of the difference problem (3.1) in L p,τ (E) obey the almost coercive inequality where M 1 is independent not only of ϕ τ , but also of τ.
Theorem 3.8.Let A is a strongly positive operator in a Banach space E with spectral angle φ(A,E) < π/2l.Then the solutions of the difference problem (3.1) in C α (τ,E) and C(τ,E α ), 10 Well-posedness of the difference schemes 0 < α < 1 obey the coercive inequality where M 1 is independent not only of ϕ τ , α, but also of τ.

Application
We consider 2m-order multidimensional elliptic equation where a r (x) and f (y,x) are given sufficiently smooth functions and α r (x) > 0, δ > 0 is the sufficiently large number.Here r ∈ R n is a vector with nonnegative integer components, Now, the abstract theorems given from above are applied in the investigation of difference schemes of higher order of accuracy with respect to the set all variables for approximate solution of the differential equation (4.1).The discretization of problem (4.1) is carried out in two steps.In the first step let us give the difference operator A x h by the formula The coefficients are chosen in such a way that the operator A x h approximates in a specified way the operator (see [2]) We will assume that for |ξ k h| ≤ π the symbol A(ξh,h) of the operator A x h − δ satisfies the inequalities With the help of A x h we arrive at the infinite system of ordinary differential equations A. Ashyralyev and P. E. Sobolevskiȋ 11 In the second step we replace differential equation (4.5) by the difference scheme Let us give a number of corollaries of the abstract theorems given in the above.
Theorem 4.1.Let τ and h be a sufficiently small numbers.Then the solutions of the difference schemes (4.6) satisfy the following stability estimates: where M does not depend on {ϕ k h } ∞ −∞ , α, β, h and τ.The proof of Theorem 4.1 is based on the abstract Theorem 3.4 and the positivity of the operator A x h in C β h [6].Theorem 4.2.Let τ and h be a sufficiently small numbers.Then the solutions of the difference schemes (4.6) satisfy the following almost coercive stability estimates:  are equivalent uniformly in h [6] and on the following theorem on the structure of the fractional spaces E α ((A x h ) 1/2 ,C h ).
where M(α,β) does not depend on {ϕk h } ∞ −∞ ,h and τ.The proof of Theorem 4.3 is based on the abstract Theorem 3.8 and the positivity of the operator A x h in C β h and the coercivity inequality for an elliptic operator A x h in C β h , 0 < β < 1 and on the fact that for any 0 < β < 1/2m the norms in the spaces E β (A x h ,C h ) and C 2mβ h The proof of Theorem 4.2 is based on the abstract Theorem 3.7, the positivity of the operator A x h in C h and on the almost coercivity inequality for an elliptic operator A x h in C h and on the estimate Theorem 4.3.Let τ and h be a sufficiently small numbers.Then the solutions of the difference schemes (4.6) satisfy the coercivity estimates: where M does not depend on {ϕ k h } ∞ −∞ , h and τ.