关于伽辽金方法收敛性的判别准则

<正> 伽辽金方法的重要性已为工程数学界所公认.有关它的收敛性的讨论,亦有大量文献与专著.但从算子方程的角度来看,所加的条件还很苛刻.本文则在较一般的条件下给出了伽辽金方法收敛性的一系列判别准则.我们相信,这些结果对于实际应用将是有益的.

ON THE CONVERGENCE CRITERION OF GALERKIN METHOD

Let H be a separable Hilbert space and A .D(f) → R(f) be a linear closed operator with D(f) =R(f) = H. We consider an independent sequence φ_j ∈ D(A), j = 1, 2,… Sequence Aφ_j, j =1, 2,…, is total in H. Let H_n be the subspace spanned by φ_1…,φ_n, and E_m :H →H_n be the projection mapping,and put A_n = E_nAE_n.Suppose the element f ∈ H and the functional equation Ax = f(1)are given. To replace the solution of Eq. (1), we consider the algebraic equation A_nX = E_nf. (2)The solution of Eq. (2) is cal]ed the Galerkin approximate solution. In this paper, the author gives the necessary and sufficient conditions and sufficient conditons for the Galerkin approximate solution to converge to the exact solution. The main results are as follows :1. The Galerkin approximate solutions x_n ∈H_n are convergent for n → ∞, iff there exists a constant ν > 0 such that where ν is independent of n. If the condition (3) is satisfied, and A~(-1) exists, then for f ∈H, Eq. (1) has the unique solution to which the Galerkin approximate solution converges. Moreover, for error estimate we have p. If there exists a positive constant ν > 0 such that then the Galerkin approximate solution covenges to the exact solution.