The best approximation method in computational mechanics
With the overwhelming use of computers in engineering, science and physics, the approximate solution of complex mathematical systems of equations is almost commonplace. The Best Approximation Method unifies many of the numerical methods used in computational mechanics. Nevertheless, despite the vast quantities of synthetic data there is still some doubt concerning the validity and accuracy of these approximations. This publication assists the computer modeller in his search for the best approximation by presenting functional analysis concepts. Computer programs are provided which can be used by readers with FORTRAN capability. The classes of problems examined include engineering applications, applied mathematics, numerical analysis and computational mechanics. The Best Approximation Method in Computational Mechanics serves as an introduction to functional analysis and mathematical analysis of computer modelling algorithms. It makes computer modellers aware of already established principles and results assembled in functional analysis.
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Topics in Functional Analysis
Hilbert Space and Generalized Fourier Series
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analytic functions approximation error approximative boundary auxilliary conditions basis elements basis functions Bessel's inequality Best Approximation Method boundary conditions boundary value Cauchy sequence column vectors COMMON/BLK computer program Consider CONTINUE CVBEM approximation defined DEFINITION differential equation dimension dot product evaluation points exact solution example problems exists finite follows Fourier coefficients Fourier series given Gm(C Gram-Schmidt Hilbert space Ideal Fluid Flow implies inner product space INPUT DATA interior interval Ker(L Laplace equation Lebesgue integral linear operator equation linearly independent Lp(S matrix system measurable set metric minimized modeling NODAL POINT nonzero rows notation noted orthonormalized Poisson Problem polynomial problem boundary problem domain READ(NRD real linear space real numbers relative error row echelon form satisfying set of basis sinx solving stream function subspace trial functions two-dimensional variable VECTOR EXPANSION vector representations vector space weighting factor