A. Dzhishkariani

A. Razmadze Mathematical Institute, Georgian Academy of Sciences Tbilisi, Georgia

PROJECTIVE METHODS FOR THE ELLIPTIC BOUNDARY VALUE PROBLEMS

Projective methods originate in Ritz’s (1908) and Galerkin’s (1915) works. These methods have been developed in the Friedrichs, Kantorovich, Michlin, Vainikho, Marchuk, Agoshkov, Siarle, Strang, Fix and others works.

In 1963, we obtained the general estimate for the equation Au = f, u ∈ D(A), f ∈ H,

∥u − u_{n}∥_{L_{2}(Ω)} ≤ Cλ_{n+1}^{−1}∥Au_{n} − f∥,

A is a self-adjoint positive definite operator in the Hilbert space H ≡ L_{2}, 0 < λ_{1} ≤ ⋯ ≤ λ_{n} → ∞, n → ∞, Bφ_{k} = λ_{k}φ_{k}, k = 1, 2, …, D(B) = D(A), ∥Au_{n} − f∥ → 0, n → ∞, φ_{k} are the basic functions.

The basic estimate of F.E.M. (Strang, Fix, Siarle, etc.) is

∥u − u_{h}∥_{H_{A}} = O(h^{k−m}),

where h is a mesh size, 2m is the order of differential operator A, u ∈ W_{2}^{k}(Ω), k − 1 is the order of finite polynomials, H_{A} is the energetic space. By us was obtained the estimate (DAN USSR, 1980) O(h^{k−s}), m < s ≤ q, when u_{h} ∈ W_{2}^{q}(Ω).

In 1993, Porter and Stirling proposed projective-iterative scheme (cyclic) for the second kind equation u + T u = f, u, f ∈ H.

This scheme for the equation Au + Ku = f, u ∈ D(A), f ∈ H, is generalized by us, when the form of inverse operator A is known. The estimates

∥u − u_{h,l}∥_{L_{2}(Ω)} = O(h^{k+m(l+1)}),
∥u − u_{h,l}∥_{C(Ω)} = O(h^{k+m(l+1)− \frac{p}{2}} ), u_{m} > p, k ≥ 2m,

where u ∈ W_{2}^{k}(Ω), l is a number of cycle, p is a dimension of the domain Ω, are obtained.