Approximation on a rectangular grid : with application to finite element methods and other problems

The present monograph has points in common with two branches of analysis. One of them is the variational-difference method (the finite element method), the other is the constructive theory of functions. The starting point is the construction of special classes of coordinate functions for the variational-difference method. It is based on elementary transformations .of the independent variables of given "primitive" functions. After the construction of the coordinate functions, the next step is to approximate functions of a given class by linear combinations of the coordinate functions, and to derive in some appropriate norm an estimate of the error. Clearly, this is a problem closely connected with the constructive theory of functions. The monograph contains 11 chapters. Chapter I discusses Courant's basic idea which is central to the construction of variational-difference methods. One of Courant's examples, from which the notion of a primitive function follows naturally, is examined in some detail. The general definition of a primitive function and the method of construction for the corresponding coordinate functions are given and discussed. Chapters II-VI are more closely connected with the constructive theory of functions. The completeness of the coordinate systems defined in Chapter I are studied, as well as the order of approximation obtained through the use of linear combinations of these functions. Their completeness in Sobolev spaces are examined in Chapter II, while related orders of approximation are derived in Chapter III.

I: The Primitive Functions.- §1. The Variational-Difference Method.- §2. An Example.- §3. The Basic Properties of Variational-Difference Matrices.- §4. Primitive Functions and Coordinate Functions.- §5. Interpolatory Properties of Primitive Systems of Functions.- II: Completeness and Fundamental Completeness Conditions.- §1. Approximation of Smooth Functions.- §2. Extensions of Functions.- §3. Completeness in Sobolev Spaces.- §4. On the Minimum Number of Primitive Functions.- §5. The Necessity of the Fundamental Completeness Conditions.- §6. One-Dimensional Primitive Systems.- §7. Primitive Systems of Higher Dimensions with Zero Degree.- §8. Primitive Systems with m = s = 2.- §9. Product Primitive Systems.- III: Order of Approximation.- §1. Order of Approximation using the Uniform Norm.- §2. On the Averaging of Functions.- §3. The Order of Approximation for Sobolev Spaces.- §4. Estimation of the Constants for the Simplest Case.- §5. Approximation Using Product Primitive Functions.- §6. Strengthened Fundamental Completeness Conditions.- §7. Some General Considerations.- §8. A More General Class of Primitive Systems.- IV: Primitive Functions with Wide Support.- §1. Definitions.- §2. Fundamental Completeness Conditions for One-Dimensional Systems.- §3. Example: The Parabolic Approximation.- §4. Fundamental Completeness Conditions for Systems of Arbitrary Dimension.- V: Approximation in One-Dimensional Degenerate Norms.- §1. The Formulation of the Problem.- §2. On the Completeness of Coordinate Systems Which are Complete with Respect to Non-Degenerate Norms.- §3. Equations of Second Order with Weak Degeneracy.- §4. The Case 1 ? ? ? 2.- §5. Properties of the Solution.- §6. Improved Estimates.- §7. The Case ? ? 2.- §8. More General Equations.- §9. Approximation in L2.- §10. Other Boundary Conditions.- VI: Some Degenerate Two-Dimensional Norms.- §1. Approximations for Radially-Symmetric Grids.- §2. Estimation of the First Integral.- §3. Estimation of the Second Integral.- §4. The Class C(2,?).- §5. Approximation on Lp and C.- §6. Degenerate Second Order Elliptic Equations.- VII: Approximation of Eigenvalues.- §1. On the Order of the Largest Approximate Eigenvalue. Formulation of the Problem.- §2. The Rayleigh-Ritz Process.- §3. One-Dimensional Variational-Difference Processes.- §4. The Case of Several Variables.- §5. Error Estimate for Fixed Eigenvalues.- VIII: Construction of Variational Difference Equations.- §1. First Boundary Value Problems: Equations with Constant Coefficients on a Cube.- §2. First Boundary Value Problems: Equations with Variable Coefficients on a Cube.- §3. First Boundary Value Problems: Natural Boundary Conditions.- §4. First Boundary Value Problems: Approximation of the Boundary Conditions.- §5. Variational-Difference Methods on an Axial-Symmetric Grid.- §6. Variational-Difference Schemes Containing a Boundary Layer: The One-Dimensional Situation.- §7. Variational-Difference Schemes Containing a Boundary Layer: The Multidimensional Situation.- §8. Non-Self Adjoint Problems.- IX: Error Estimates for the Variational-Difference Method.- §1. On the Stability of Numerical Processes.- §2. The Stability of Variational-Difference Processes — The One-Dimensional Problem.- §3. The Stability of Variational-Difference Processes — Multi-Dimensional Problems.- §4. The Stability of Variational-Difference Processes — Eigenvalue Problems.- §5. On the Condition Number of the Variational-Difference Matrix.- §6. The Case of Arbitrary Domains and Arbitrary Boundary Conditions.- §7. Numerical Example — A Degenerate Second Order Ordinary Differential Equation.- X: The Euler-Maclaurin Sum Formula.- §1. A New Derivation of the Euler-Maclaurin Sum Formula.- §2. A Related Euler-Maclaurin Sum Formula.- §3. An Euler-Maclaurin Sum Formula for the Multidimensional Cube.- §4. Integration Over a Ball.- XI: On Integral Equations.- §1. Approximation of the Kernel and Resolvent.- §2. The Accuracy of the Approximation.- §3. Rounding Error Accumulation — Absolute Estimates.- §4. Rounding Error Accumulation — Probabilistic Estimates.- §5. Integral Equations Which Can be Solved by Iteration.- §6. Some Additional Notes.- §7. Equations with Weak Singularities.- §8. Integral Equations of Heat Conduction.- References.