Integral Equations

Lemma 8 The projection method converges if and only if there exists an index N and a positiveconstant M such that for all n ≥ N, the finite-dimensional operators A n = P n A : X n → Y n areinvertible and operators A −1n P n A : X → X are uniformly bounded,The error estimate is valid‖A −1n P n A‖ ≤ M. (377)‖ϕ n − ϕ‖ ≤ (1 + M) infψ∈X n‖ψ − ϕ‖. (378)Relationship (378) is usually called the quasioptimal estimate and corresponds to the errorestimate of a projection method based on the approximation of elements from X by elementsof subspaces X n . It shows that the error of the projection method is determined by the qualityof approximation of the exact solution by elements of subspace X n .Consider the equationSϕ + Kϕ = f (379)and the projection methodsandwith subspaces X n and projection operators P n : Y → Y n .P n Sϕ n = P n f (380)P n (S + K)ϕ n = P n f (381)Lemma 9 Assume that S : X → Y is a bounded operator with a bounded inverse S −1 : Y → Xand that the projection method (380) is convergent for S. Let K : X → Y be a compact boundedoperator and S + K is injective. Then, the projection method (381) converges for S + K.Lemma 10 Assume that Y n = S(X n ) and ‖P n K − K‖ → 0, n → ∞. Then, for sufficientlylarge n, approximate equation (381) is uniquely solvable and the error estimate‖ϕ n − ϕ‖ ≤ M‖P n Sϕ − Sϕ‖is valid, where M is a positive constant depending on S and K.For operator equations in Hilbert spaces, the projection method employing an orthogonalprojection into finite-dimensional subspaces is called the Galerkin method.Consider a pair of Hilbert spaces X and Y and assume that A : X → Y is an injectiveoperator. Let X n ⊂ X and Y n ⊂ Y be the subspaces with dim X n = dim Y n = n. Then, ϕ n ∈ X nis a solution of the equation Aϕ = f obtained by the projection method generated by X n andoperators of orthogonal projections P n : Y → Y n if and only if(Aϕ n , g) = (f, g) (382)for all g ∈ Y n . Indeed, equations (382) are equivalent to P n (Aϕ n − f).Equation (382) is called the Galerkin equation.80