| Introduction |
| Aim and Scope / 1： |
| Neumann Problems / 2： |
| Introduction of Internal Approximations / 3： |
| Properties of Internal Approximations / 4： |
| Stability, Optimal Stability, and Regularity of the Convergence / 5： |
| The Case of Operators Mapping a Hilbert Space onto Its Dual / 6： |
| Finite-Element Approximations of Sobolev Spaces / 7： |
| Approximation of Nonhomogeneous Neumann Problems / 8： |
| Approximations of Nonhomogeneous Dirichlet Problems / 9： |
| A Posteriori Error Estimates / 10： |
| External and Partial Approximations / 11： |
| General Outline / 12： |
| Approximation of Solutions of Neumann Problems for Second-Order Linear Differential Equations |
| Weak Solutions of Neumann Problems for Second-Order Linear Differential Operators |
| The Neumann Boundary-Value Problem / 1-1： |
| Definition of Distributions / 1-2： |
| Weak Derivatives of a Distribution / 1-3： |
| Variational Formulation of the Problem / 1-4： |
| Weak Solutions of the Neumann Boundary-Value Problem / 1-5： |
| Sobolev Spaces / 1-6： |
| The Lax-Milgram Theorem / 1-7： |
| Approximation of an Abstract Variational Problem |
| The Galerkin Approximation of a Separable Hilbert Space / 2-1： |
| Approximation of a Hilbert Space / 2-2： |
| Internal Approximation of a Variational Equation / 2-3： |
| Existence, Uniqueness, and Convergence Properties / 2-4： |
| Estimates of Global Error / 2-5： |
| What Kind of Approximations Should Be Chosen? / 2-6： |
| Examples of Approximations of Sobolev Spaces |
| Piecewise-Linear Approximations of the Sobolev Space H[superscript 1] (I) / 3-1： |
| Estimates of Error Functions of Piecewise-Linear Approximations / 3-2： |
| Examples of Approximate Equations |
| Construction of a Finite-Difference Scheme / 4-1： |
| A Simpler Finite-Difference Scheme / 4-2： |
| Approximations of Hilbert Spaces |
| Hilbert Spaces and Their Duals |
| Dual of a Hilbert Space and Canonical Isometry |
| Example: Finite-Dimensional Hilbert Spaces |
| Hahn-Banach Theorem |
| Dual of a Dense Subspace |
| Imbedding of a Space into Its Dual |
| Example: Imbedding of Spaces of Functions into Spaces of Distributions |
| Dual of Closed Subspaces and Factor Spaces |
| Applications to Error Estimates / 1-8： |
| Dual of a Product / 1-9： |
| Dual of Domains of Operators / 1-10： |
| Examples: Dual of Sobolev Spaces H[subscript 0 superscript m](I) / 1-11： |
| Properties of Bounded Sets of Operators; Uniform Boundedness / 1-12： |
| Banach Theorem / 1-13： |
| Dual of Sobolev Spaces H[superscript m](I) / 1-14： |
| The Riesz-Fredholm Alternative / 1-15： |
| V-Elliptic and Coercive Operators / 1-16： |
| Quasi-Optimal Approximations |
| Stability Functions |
| Duality Relations between Error and Stability Functions |
| Estimates of the Stability Functions |
| Quasi-Optimal Approximations; Estimate of the Error Function |
| Truncation Errors and Error Functions |
| Optimal Approximations |
| Eigenvalues and Eigenvectors of Symmetric Compact Operators |
| Optimal Galerkin Approximations |
| Convergence and Optimality Properties / 3-3： |
| Spaces H[subscript Theta] / 3-4： |
| Optimal Restrictions and Prolongations; Applications |
| Optimal Restrictions and Prolongations |
| Dual Approximations |
| Construction of Optimal Prolongations and Restrictions / 4-3： |
| Miscellaneous Remarks / 4-4： |
| Characterization of Error and Stability Functions / 4-5： |
| Spaces of Order [Theta] / 4-6： |
| Approximation of Operators |
| Internal Approximations |
| Construction of an Internal Approximate Equation |
| The Case of Finite-Dimensional Discrete Spaces |
| The Case of Operators from V onto V[prime] |
| Stability of Internal Approximations of Operators |
| Convergence and Error Estimates |
| Approximation of a Sum of an Isomorphism and a Compact Operator |
| Approximation of Coercive and V-Elliptic Operators |
| Optimal and Quasi-Optimal Stability |
| Regularity of the Convergence and Estimates of Error in Terms of n-Width |
| Stability and Convergence in Smaller Spaces |
| Stability and Convergence in Larger Spaces |
| Approximation of the Value of a Functional at a Solution |
| Discrete Convergence, Consistency, and Optimal Approximation of Linear Operators |
| Discrete Convergence and Consistency |
| Optimal Approximation of Operators and Internal Approximations |
| Estimates of Error and Discrete Errors |
| Finite-Element Approximation of Functions of One Variable |
| Approximation of Functions of L[superscript 2] by Step Functions and by Convolution |
| The Space L[superscript 2] and the Discrete Space L[subscript h superscript 2]] |
| The Prolongations P[subscript h superscript 0] |
| The Restrictions r[subscript h] |
| The Theorem of Convergence |
| Convolution of Functions and Measures |
| Approximation by Convolution |
| Piecewise-Polynomial Approximations of Sobolev Spaces H[superscript m] |
| Finite-Difference Operators |
| Construction of Approximations of the Space H[superscript m] |
| Convergence Theorem |
| Explicit Form of Functions [Pi subscript m] |
| Properties of the Prolongations p[subscript h superscript m] |
| Optimal Properties of Prolongations p[subscript h superscript m] / 2-7： |
| Finite-Element Approximations of Sobolev Spaces H[superscript m] |
| Finite-Element Approximations |
| The Criterion of m-Convergence |
| Characterization of Convergent Finite-Element Approximations |
| Stability Properties of Finite-Element Approximations |
| Finite-Element Approximation of Functions of Several Variables |
| Approximations of the Sobolev Spaces H[superscript m](R[superscript n]) |
| Notations |
| (2m + 1)[superscript n]-Level Piecewise-Polynomial Approximations |
| [2(2m)[superscript n] - (2m - 1)[superscript n]]-Level Piecewise-Polynomial Approximations |
| Approximations of the Sobolev Spaces H[superscript m]([Omega]) |
| Sobolev Spaces H[superscript m]([Omega]) |
| Finite-Element Approximations of H[superscript m]([Omega]) |
| Quasi-Optimal Finite-Element Approximations of H[superscript m]([Omega]) |
| Piecewise-Polynomial Approximations of H[superscript m]([Omega]) |
| Approximation of the Sobolev Spaces H[subscript 0 superscript m]([Omega]) |
| Sobolev Spaces H[subscript 0 superscript m]([Omega]) |
| Finite-Element Approximations of H[subscript 0 superscript m]([Omega]) |
| Convergent Finite-Element Approximations of H[subscript 0 superscript m]([Omega]) |
| Boundary-Value Problems and the Trace Theorem |
| Some Variational Boundary-Value Problems for the Laplacian |
| The Laplacian |
| Characterization of Sobolev Spaces H[subscript 0 superscript 1]([Omega]) |
| The Green Formula |
| The Dirichlet Problem for the Laplacian |
| The Neumann Problem for the Laplacian |
| A Mixed Problem for the Laplacian |
| An Oblique Problem for the Laplacian |
| Existence and Uniqueness of the Solutions |
| Variational Boundary-Value Problems and Their Adjoints |
| Spaces V, H and Operator [gamma] |
| Formal Operator [Lambda] Associated with a(u, v) |
| Abstract Neumann and Dirichlet Problems Associated with a(u, v) |
| Mixed Type Boundary-Value Problems Associated with a(u, v) |
| Existence and Uniqueness of the Solutions of Boundary-Value Problems |
| Formal Adjoint of an Operator and Green's Formula |
| Theorems of Regularity / 2-8： |
| The Trace Theorem and Properties of Sobolev Spaces |
| Statement of the Trace Theorem |
| Change of Coordinates |
| Sobolev Spaces H[superscript s](R[superscript n]) for Real Numbers s |
| Sobolev Spaces H[superscript s]([Gamma] and H[superscript s]([Omega]) |
| Trace Operators and Operators of Extension: Theorems of Density / 3-5： |
| Properties of the Spaces H[superscript m](R[subscript + superscript n]) / 3-6： |
| Proof of the Trace Theorem / 3-7： |
| Sobolev Inequalities and the Trace Theorem in Space H[superscript s]([Omega]) / 3-8： |
| Theorem of Compactness / 3-9： |
| Examples of Boundary-Value Problems |
| Boundary-Value Problems for Second-Order Differential Operators |
| Second-Order Linear Differential Operators |
| Elliptic Second-Order Partial Differential Operators |
| The Dirichlet Problem |
| The Neumann Problem |
| Mixed Problems |
| Oblique Problems |
| Interface Problems |
| The Regularity Theorem |
| Theorems of Isomorphism |
| Value of the Solution at a Point of the Boundary |
| Problems with Elliptic Differential Boundary Conditions |
| Boundary-Value Problems for Differential Operators of Higher Order |
| Linear Differential Operators of Order 2k |
| Regularity and Theorems of Isomorphism |
| Other Boundary-Value Problems |
| Boundary Value Problems for [Delta][superscript 2] + [lambda] |
| Approximation of Neumann-Type Problems |
| Theorems of Convergence and Error Estimates |
| Internal Approximation of a Neumann-type Problem |
| Convergence and Estimates of Error in Larger Spaces |
| Approximation of Neumann Problems for Elliptic Operators of Order 2k |
| Approximation of Neumann Problems for Elliptic Differential Operators |
| Convergence Properties of Finite Element Approximations of Neumann Problems |
| The (2m + 1)[superscript n]-Level Approximations of the Neumann Problem |
| The [2(2m)[superscript n] - (2m - 1)[superscript n]]-Level Approximations of the Neumann Problem |
| Approximations of the Spaces H[superscript k]([Omega], [Lambda] and H([Omega], [Lambda] |
| Approximation of Other Neumann-Type Problems |
| Approximation of the Value of the Solution at a Point of the Boundary |
| Approximation of Oblique Boundary-Value Problems |
| Approximation of a Problem with Elliptic Boundary Conditions |
| Approximation of Interface Problems |
| Approximation of the Neumann Problem for [Delta][superscript 2] + [gamma] |
| Perturbed Approximations and Least-Squares Approximations |
| Perturbed Approximations |
| Internal Approximation of a Variational Boundary-Value Problem |
| Perturbed Approximation of a Variational Boundary-Value Problem |
| Convergence in the Initial Space |
| Estimates of Error |
| Convergence in Smaller Spaces |
| Convergence in Larger Spaces |
| Perturbed Approximations of Boundary-Value Problems |
| Perturbed Approximations by Finite-Element Approximations |
| Error Estimates and Regularity of the Convergence |
| The 3[superscript n]-level Perturbed Approximation of the Dirichlet Problem |
| Least-Squares Approximations |
| Least-Squares Approximation Schemes |
| Error Estimates (I) |
| Error Estimates (II) |
| Least-Squares Approximations of Dirichlet Problems |
| Conjugate Problems and A Posteriori Error Estimates |
| Conjugate Problems of Boundary-Value Problems |
| First Example of a Conjugate Problem |
| Second Example of a Conjugate Problem |
| Construction of Conjugate Problems |
| Applications to the Approximation of Dirichlet Problems |
| Approximation of the Dirichlet Problem (I) |
| Approximation of the Dirichlet Problem (II) |
| The Case of Second-Order Differential Operators |
| Finite-Element Approximations of the Spaces H[superscript k]([Omega], D*) |
| Spaces H[superscript k]([Omega], D*) |
| Approximations of the Space H[superscript k]([Omega], D*) |
| Approximation of the Second Example of a Conjugate Problem |
| Approximation of the Conjugate Dirichlet Problem |
| Properties of the Discrete Conjugate Problem |
| External Approximations; Stability, Convergence, and Error Estimates |
| Definition of External Approximations |
| Example: Partial Approximations of a Finite Intersection of Spaces |
| Stability and Convergence of External Approximations of Operators |
| Estimates of Error and Regularity of the Convergence |
| Properties of the External Error Functions |
| External and Partial Approximations of Variational Equations |
| Partial Approximation of a Split Variational Equation |
| External Approximation of Variational Equations |
| Partial Approximation of Neumann Problems |
| Perturbed Partial Approximation of Boundary-Value Problems |
| Partial Approximations of Sobolev Spaces |
| Spaces H([Omega], D[subscript i]) |
| Partial Approximations of the Sobolev Space H[superscript 1]([Omega]) |
| Estimates of Truncation Errors and External Error Functions |
| Partial Approximations of the Sobolev Spaces H[superscript m]([Omega]) and H[subscript 0 superscript m]([Omega]) |
| Partial Approximation of Boundary-Value Problems |
| Partial Approximation of Second-Order Linear Operators |
| Partial Approximation of the Neumann Problem |
| Perturbed Partial Approximation of Mixed Boundary-Value Problems |
| Estimates of Error in the Interior |
| Partial Approximations of Higher-Order Differential Operators |
| Comments |
| References |
| Index |
図書
