Approximation of Elliptic Boundary-Value Problems

Overview


A marriage of the finite-differences method with variational methods for solving boundary-value problems, the finite-element method is superior in many ways to finite-differences alone. This self-contained text for advanced undergraduates and graduate students is intended to imbed  this combination of methods into the framework of functional analysis and to explain its applications to approximation of nonhomogeneous boundary-value problems for elliptic ...
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Overview


A marriage of the finite-differences method with variational methods for solving boundary-value problems, the finite-element method is superior in many ways to finite-differences alone. This self-contained text for advanced undergraduates and graduate students is intended to imbed  this combination of methods into the framework of functional analysis and to explain its applications to approximation of nonhomogeneous boundary-value problems for elliptic operators.
The treatment begins with a summary of the main results established in the book. Chapter 1 introduces the variational method and the finite-difference method in the simple case of second-order differential equations. Chapters 2 and 3 concern abstract approximations of Hilbert spaces and linear operators, and Chapters 4 and 5 study finite-element approximations of Sobolev spaces. The remaining four chapters consider several methods for approximating nonhomogeneous boundary-value problems for elliptic operators.
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Product Details

  • ISBN-13: 9780486457918
  • Publisher: Dover Publications
  • Publication date: 2/27/2007
  • Series: Dover Books on Mathematics Series
  • Pages: 384
  • Product dimensions: 5.30 (w) x 8.40 (h) x 0.80 (d)

Table of Contents


Introduction     1
Aim and Scope     1
Neumann Problems     2
Introduction of Internal Approximations     5
Properties of Internal Approximations     7
Stability, Optimal Stability, and Regularity of the Convergence     9
The Case of Operators Mapping a Hilbert Space onto Its Dual     10
Finite-Element Approximations of Sobolev Spaces     12
Approximation of Nonhomogeneous Neumann Problems     14
Approximations of Nonhomogeneous Dirichlet Problems     16
A Posteriori Error Estimates     20
External and Partial Approximations     21
General Outline     24
Approximation of Solutions of Neumann Problems for Second-Order Linear Differential Equations     25
Weak Solutions of Neumann Problems for Second-Order Linear Differential Operators     25
The Neumann Boundary-Value Problem     25
Definition of Distributions     26
Weak Derivatives of a Distribution     26
Variational Formulation of the Problem     27
Weak Solutions of the Neumann Boundary-Value Problem     27
Sobolev Spaces     29
The Lax-Milgram Theorem     30
Approximation of an Abstract Variational Problem     31
The Galerkin Approximation of a Separable Hilbert Space     31
Approximation of a Hilbert Space     32
Internal Approximation of a Variational Equation     33
Existence, Uniqueness, and Convergence Properties     34
Estimates of Global Error     35
What Kind of Approximations Should Be Chosen?     36
Examples of Approximations of Sobolev Spaces     37
Piecewise-Linear Approximations of the Sobolev Space H[superscript 1] (I)     37
Estimates of Error Functions of Piecewise-Linear Approximations     38
Examples of Approximate Equations     40
Construction of a Finite-Difference Scheme     40
A Simpler Finite-Difference Scheme     43
Approximations of Hilbert Spaces     45
Hilbert Spaces and Their Duals     45
Dual of a Hilbert Space and Canonical Isometry     46
Example: Finite-Dimensional Hilbert Spaces     47
Hahn-Banach Theorem     47
Dual of a Dense Subspace     48
Imbedding of a Space into Its Dual     49
Example: Imbedding of Spaces of Functions into Spaces of Distributions     50
Dual of Closed Subspaces and Factor Spaces     51
Applications to Error Estimates     52
Dual of a Product     53
Dual of Domains of Operators     54
Examples: Dual of Sobolev Spaces H[subscript 0 superscript m](I)     55
Properties of Bounded Sets of Operators; Uniform Boundedness     56
Banach Theorem     57
Dual of Sobolev Spaces H[superscript m](I)     58
The Riesz-Fredholm Alternative     60
V-Elliptic and Coercive Operators     60
Quasi-Optimal Approximations     62
Stability Functions     63
Duality Relations between Error and Stability Functions     63
Estimates of the Stability Functions     64
Quasi-Optimal Approximations; Estimate of the Error Function     65
Truncation Errors and Error Functions     66
Optimal Approximations     66
Eigenvalues and Eigenvectors of Symmetric Compact Operators     67
Optimal Galerkin Approximations     68
Convergence and Optimality Properties     69
Spaces H[subscript Theta]     70
Optimal Restrictions and Prolongations; Applications     72
Optimal Restrictions and Prolongations     73
Dual Approximations     74
Construction of Optimal Prolongations and Restrictions     75
Miscellaneous Remarks      76
Characterization of Error and Stability Functions     78
Spaces of Order [Theta]     80
Approximation of Operators     82
Internal Approximations     82
Construction of an Internal Approximate Equation     83
The Case of Finite-Dimensional Discrete Spaces     84
The Case of Operators from V onto V[prime]     84
Stability of Internal Approximations of Operators     85
Convergence and Error Estimates     86
Approximation of a Sum of an Isomorphism and a Compact Operator     88
Approximation of Coercive and V-Elliptic Operators     90
Optimal and Quasi-Optimal Stability     92
Regularity of the Convergence and Estimates of Error in Terms of n-Width     95
Stability and Convergence in Smaller Spaces     95
Stability and Convergence in Larger Spaces     98
Approximation of the Value of a Functional at a Solution     101
Discrete Convergence, Consistency, and Optimal Approximation of Linear Operators     102
Discrete Convergence and Consistency     103
Optimal Approximation of Operators and Internal Approximations     106
Estimates of Error and Discrete Errors     107
Finite-Element Approximation of Functions of One Variable      109
Approximation of Functions of L[superscript 2] by Step Functions and by Convolution     109
The Space L[superscript 2] and the Discrete Space L[subscript h superscript 2]]     110
The Prolongations P[subscript h superscript 0]     110
The Restrictions r[subscript h]     110
The Theorem of Convergence     111
Convolution of Functions and Measures     112
Approximation by Convolution     115
Piecewise-Polynomial Approximations of Sobolev Spaces H[superscript m]     116
Finite-Difference Operators     116
Construction of Approximations of the Space H[superscript m]     117
Convergence Theorem     118
Explicit Form of Functions [Pi subscript m]     119
Properties of the Prolongations p[subscript h superscript m]     121
Estimates of the Stability Functions     123
Optimal Properties of Prolongations p[subscript h superscript m]     124
Finite-Element Approximations of Sobolev Spaces H[superscript m]     124
Finite-Element Approximations     125
The Criterion of m-Convergence     128
Characterization of Convergent Finite-Element Approximations     130
Stability Properties of Finite-Element Approximations     133
Finite-Element Approximation of Functions of Several Variables     135
Approximations of the Sobolev Spaces H[superscript m](R[superscript n])     136
Notations     136
Finite-Element Approximations     138
(2m + 1)[superscript n]-Level Piecewise-Polynomial Approximations     141
[2(2m)[superscript n] - (2m - 1)[superscript n]]-Level Piecewise-Polynomial Approximations     142
Approximations of the Sobolev Spaces H[superscript m]([Omega])     149
Sobolev Spaces H[superscript m]([Omega])     149
Finite-Element Approximations of H[superscript m]([Omega])     150
Quasi-Optimal Finite-Element Approximations of H[superscript m]([Omega])     151
Piecewise-Polynomial Approximations of H[superscript m]([Omega])     154
Approximation of the Sobolev Spaces H[subscript 0 superscript m]([Omega])     157
Sobolev Spaces H[subscript 0 superscript m]([Omega])     157
Finite-Element Approximations of H[subscript 0 superscript m]([Omega])     159
Convergent Finite-Element Approximations of H[subscript 0 superscript m]([Omega])     159
Boundary-Value Problems and the Trace Theorem     162
Some Variational Boundary-Value Problems for the Laplacian     162
The Laplacian     163
Characterization of Sobolev Spaces H[subscript 0 superscript 1]([Omega])     163
The Green Formula     164
The Dirichlet Problem for the Laplacian     165
The Neumann Problem for the Laplacian     166
A Mixed Problem for the Laplacian     167
An Oblique Problem for the Laplacian     168
Existence and Uniqueness of the Solutions     170
Variational Boundary-Value Problems and Their Adjoints     170
Spaces V, H and Operator [gamma]     171
Formal Operator [Lambda] Associated with a(u, v)     172
The Green Formula     172
Abstract Neumann and Dirichlet Problems Associated with a(u, v)     174
Mixed Type Boundary-Value Problems Associated with a(u, v)     175
Existence and Uniqueness of the Solutions of Boundary-Value Problems     178
Formal Adjoint of an Operator and Green's Formula     182
Theorems of Regularity     184
The Trace Theorem and Properties of Sobolev Spaces     187
Statement of the Trace Theorem     187
Change of Coordinates     188
Sobolev Spaces H[superscript s](R[superscript n]) for Real Numbers s     189
Sobolev Spaces H[superscript s]([Gamma] and H[superscript s]([Omega])     190
Trace Operators and Operators of Extension: Theorems of Density      191
Properties of the Spaces H[superscript m](R[subscript + superscript n])     194
Proof of the Trace Theorem     196
Sobolev Inequalities and the Trace Theorem in Space H[superscript s]([Omega])     198
Theorem of Compactness     199
Examples of Boundary-Value Problems     201
Boundary-Value Problems for Second-Order Differential Operators     201
Second-Order Linear Differential Operators     201
Elliptic Second-Order Partial Differential Operators     202
The Dirichlet Problem     203
The Neumann Problem     204
Mixed Problems     204
Oblique Problems     205
Interface Problems     206
The Regularity Theorem     208
Theorems of Isomorphism     211
Value of the Solution at a Point of the Boundary     212
Problems with Elliptic Differential Boundary Conditions     213
Boundary-Value Problems for Differential Operators of Higher Order     214
Linear Differential Operators of Order 2k     214
The Dirichlet Problem     215
The Neumann Problem     215
Regularity and Theorems of Isomorphism     216
Other Boundary-Value Problems     217
Boundary Value Problems for [Delta][superscript 2] + [lambda]     218
Approximation of Neumann-Type Problems     222
Theorems of Convergence and Error Estimates     222
Internal Approximation of a Neumann-type Problem     223
Convergence and Estimates of Error in Larger Spaces     225
Approximation of Neumann Problems for Elliptic Operators of Order 2k     229
Approximation of Neumann Problems for Elliptic Differential Operators     233
Convergence Properties of Finite Element Approximations of Neumann Problems     233
The (2m + 1)[superscript n]-Level Approximations of the Neumann Problem     235
The [2(2m)[superscript n] - (2m - 1)[superscript n]]-Level Approximations of the Neumann Problem     238
Approximations of the Spaces H[superscript k]([Omega], [Lambda] and H([Omega], [Lambda]     239
Approximation of Other Neumann-Type Problems     240
Approximation of the Value of the Solution at a Point of the Boundary     240
Approximation of Oblique Boundary-Value Problems     242
Approximation of a Problem with Elliptic Boundary Conditions     243
Approximation of Interface Problems     248
Approximation of the Neumann Problem for [Delta][superscript 2] + [gamma]     250
Perturbed Approximations and Least-Squares Approximations      252
Perturbed Approximations     252
Internal Approximation of a Variational Boundary-Value Problem     253
Perturbed Approximation of a Variational Boundary-Value Problem     254
Convergence in the Initial Space     255
Estimates of Error     257
Convergence in Smaller Spaces     259
Convergence in Larger Spaces     259
Perturbed Approximations of Boundary-Value Problems     261
Perturbed Approximations by Finite-Element Approximations     261
Error Estimates and Regularity of the Convergence     264
The 3[superscript n]-level Perturbed Approximation of the Dirichlet Problem     265
Least-Squares Approximations     267
Least-Squares Approximation Schemes     267
Error Estimates (I)     269
Error Estimates (II)     270
Least-Squares Approximations of Dirichlet Problems     275
Conjugate Problems and A Posteriori Error Estimates     280
Conjugate Problems of Boundary-Value Problems     280
First Example of a Conjugate Problem     280
Second Example of a Conjugate Problem     282
Construction of Conjugate Problems     285
Applications to the Approximation of Dirichlet Problems      293
Approximation of the Dirichlet Problem (I)     293
Approximation of the Dirichlet Problem (II)     295
The Case of Second-Order Differential Operators     296
Finite-Element Approximations of the Spaces H[superscript k]([Omega], D*)     297
Spaces H[superscript k]([Omega], D*)     297
Approximations of the Space H[superscript k]([Omega], D*)     298
Approximation of the Second Example of a Conjugate Problem     302
Approximation of the Conjugate Dirichlet Problem     302
Properties of the Discrete Conjugate Problem     306
External and Partial Approximations     307
External Approximations; Stability, Convergence, and Error Estimates     307
Definition of External Approximations     307
Example: Partial Approximations of a Finite Intersection of Spaces     310
Stability and Convergence of External Approximations of Operators     311
Estimates of Error and Regularity of the Convergence     312
Properties of the External Error Functions     314
External and Partial Approximations of Variational Equations     317
Partial Approximation of a Split Variational Equation     317
External Approximation of Variational Equations     319
Partial Approximation of Neumann Problems     321
Perturbed Partial Approximation of Boundary-Value Problems     324
Partial Approximations of Sobolev Spaces     328
Spaces H([Omega], D[subscript i])     328
Partial Approximations of the Sobolev Space H[superscript 1]([Omega])     329
Estimates of Truncation Errors and External Error Functions     332
Partial Approximations of the Sobolev Spaces H[superscript m]([Omega]) and H[subscript 0 superscript m]([Omega])     334
Partial Approximation of Boundary-Value Problems     336
Partial Approximation of Second-Order Linear Operators     336
Partial Approximation of the Neumann Problem     338
Perturbed Partial Approximation of Mixed Boundary-Value Problems     340
Estimates of Error in the Interior     342
Partial Approximations of Higher-Order Differential Operators     343
Comments     346
References     349
Index     355
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