The Best Approximation Method in Computational Mechanics

1 Topics in Functional Analysis.- 1.0 Introduction.- 1.1 Set Theory.- 1.2 Functions.- 1.3 Matrices.- 1.4 Solving Matrix Systems.- 1.5 Metric Spaces.- 1.6 Linear Spaces.- 1.7 Normed Linear Spaces.- 1.8 Approximations.- 2 Integration Theory.- 2.0 Introduction.- 2.1 Reimann and Lebesgue Integrals: Step and Simple Functions.- 2.2 Lebesgue Measure.- 2.3 Measurable Functions.- 2.4 The Lebesgue Integral.- 2.5 Key Theorems in Integration Theory.- 2.6 Lp Spaces.- 2.7 The Metric Space, Lp.- 2.8 Convergence of Sequences.- 2.9 Capsulation.- 3 Hilbert Space and Generalized Fourier Series.- 3.0 Introduction.- 3.1 Inner Product and Hilbert Space.- 3.2 Best Approximations in an Inner Product Space.- 3.3 Approximations in L2(E).- 3.4 Vector Representations and Best Approximations.- 3.5 Computer Program.- 4 Linear Operators.- 4.0 Introduction.- 4.1 Linear Operator Theory.- 4.2 Operator Norms.- 4.3 Examples of Linear Operators in Engineering.- 4.4 Superposition.- 5 The Best Approximation Method.- 5.0 Introduction.- 5.1 An Inner Product for the Solution of Linear Operator Equations.- 5.2 Definition of Inner Product and Norm.- 5.3 Generalized Fourier Series.- 5.4 Approximation Error Evaluation.- 5.5 The Weighted Inner Product.- 5.6 Considerations in Choosing Basis Functions.- 6 The Best Approximation Method: Applications.- 6.0 Introduction.- 6.1 Sensitivity of Computational Results to Variation in the Inner Product Weighting Factor.- 6.2 Solving Two-Dimensional Potential Problems.- 6.3 Application to Other Linear Operators.- 6.4 Computer Program: Two-Dimensional Potential Problems Using Real Variable Basis Functions.- 6.5 Application of Computer Program.- 7 Solving Potential Problems using the Best Approximation Method.- 7.0 Introduction.- 7.1 The Complex Variable Boundary Element Method.-7.2 Mathematical Development.- 7.3 The CVBEM and W?.- 7.4 The Space W?A.- 7.5 Applications.- 7.6 Computer Program: Two-Dimensional Potential Problems using Analytic Basis Functions (CVBEM).- 7.7 Modelling Groundwater Contaminant Transport.- 7.8 Three Dimensional Potential Problems.- 8 Applications to Linear Operator Equations.- 8.0 Introduction.- 8.1 Data Fit Analysis.- 8.2 Ordinary Differential Equations.- 8.3 Best Approximation of Function.- 8.4 Matrix Systems.- 8.5 Linear Partial Differential Equations.- 8.6 Linear Integral Equations.- References.- Appendix A Derivation of CVBEM Approximation Function.- Appendix B Convergence of CVBEM Approximator.- Appendix C The Approximate Boundary for Error Analysis.