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Fundamentals of the Theory of Difference Schemes
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absolutely stable algorithm applied approximate solution approximation error arbitrary assume basis functions boundary conditions boundary problem coefficients components compute conjugate gradient method consider constraints construct corresponding defined denote derivatives difference equations difference scheme differential equation Dirichlet problem domain of definition eigenvalues eigenvectors elements estimate exact solution expansion finite follows formula Fourier Galerkin method given Hilbert space identity inequality initial integral interpolation interval introduce iteration method iterative process linear algebraic mathematical physics matrix mesh minimization nonstationary norm obtain optimal parameters perturbations piecewise-linear points polynomial positive definite priori problems of mathematical rate of convergence region relation Ritz method satisfied scalar product second-order approximation Section sequence solving spectral problem spline splitting-up method sufficiently smooth suppose symmetric theorem theory tion transition operator unique values variables variational-difference well-posed well-posed problems yields