| Continuum Boundary Value Problems and the Need for Numerical Discretization. Finite Difference Methods | p. 1 |
| Introduction | p. 1 |
| Some Examples of Continuum Problems | p. 2 |
| Finite Differences in One Dimension | p. 6 |
| Derivative Boundary Conditions | p. 14 |
| Nonlinear Problems | p. 18 |
| Finite Differences in More Than One Dimension | p. 22 |
| Problems Involving Irregularly Shaped Regions | p. 30 |
| Nonlinear Problems in More Than One Dimension | p. 32 |
| Approximation and Convergence | p. 33 |
| Concluding Remarks | p. 34 |
| References | p. 36 |
| Suggested Further Reading | p. 37 |
| Weighted Residual Methods: Use of Continuous Trial Functions | p. 38 |
| Introduction-Approximation by Trial Functions | p. 38 |
| Weighted Residual Approximations | p. 42 |
| Approximation to the Solutions of Differential Equations and the Use of Trial Function-Weighted Residual Forms. Boundary Conditions Satisfied by Choice of Trial Functions | p. 49 |
| Simultaneous Approximation to the Solutions of Differential Equations and to the Boundary Conditions | p. 57 |
| Natural Boundary Conditions | p. 63 |
| Boundary Solution Methods | p. 71 |
| Systems of Differential Equations | p. 75 |
| Nonlinear Problems | p. 89 |
| Concluding Remarks | p. 93 |
| References | p. 93 |
| Suggested Further Reading | p. 94 |
| Piecewise Defined Trial Functions and the Finite Element Method | p. 95 |
| Introduction-The Finite Element Concept | p. 95 |
| Some Typical Locally Defined Narrow-Base Shape Functions | p. 96 |
| Approximation to Solutions of Differential Equations and Continuity Requirements | p. 103 |
| Weak Formulation and the Galerkin Method | p. 105 |
| Some One-Dimensional Problems | p. 106 |
| Standard Discrete System. A Physical Analogue of the Equation Assembly Process | p. 119 |
| Generalization of the Finite Element Concepts for Two- and Three-Dimensional Problems | p. 126 |
| The Finite Element Method for Two-Dimensional Heat Conduction Problems | p. 132 |
| Two-Dimensional Elastic Stress Analysis Using Triangular Elements | p. 148 |
| Are Finite Differences a Special Case of the Finite Element Method? | p. 154 |
| Concluding Remarks | p. 157 |
| References | p. 160 |
| Suggested Further Reading | p. 160 |
| Higher Order Finite Element Approximation | p. 161 |
| Introduction | p. 161 |
| Degree of Polynomial in Trial Functions and Convergence Rates | p. 162 |
| The Patch Test | p. 164 |
| Standard Higher Order Shape Functions for One-Dimensional Elements with C[superscript 0] Continuity | p. 164 |
| Hierarchical Forms of Higher Order One-Dimensional Elements with C[superscript 0] Continuity | p. 171 |
| Two-Dimensional Rectangular Finite Element Shape Functions of Higher Order | p. 178 |
| Two-Dimensional Shape Functions for Triangles | p. 185 |
| Three-Dimensional Shape Functions | p. 190 |
| Concluding Remarks | p. 190 |
| References | p. 192 |
| Suggested Further Reading | p. 192 |
| Mapping and Numerical Integration | p. 193 |
| The Concept of Mapping | p. 193 |
| Numerical Integration | p. 206 |
| More on Mapping | p. 214 |
| Mesh Generation and Concluding Remarks | p. 228 |
| References | p. 229 |
| Suggested Further Reading | p. 230 |
| Variational Methods | p. 231 |
| Introduction | p. 231 |
| Variational Principles | p. 232 |
| The Establishment of Natural Variational Principles | p. 236 |
| Approximate Solution of Differential Equations by the Rayleigh-Ritz Method | p. 244 |
| The Use of Lagrange Multipliers | p. 248 |
| General Variational Principles | p. 254 |
| Penalty Functions | p. 256 |
| Least-Squares Method | p. 259 |
| Concluding Remarks | p. 264 |
| References | p. 265 |
| Suggested Further Reading | p. 265 |
| Partial Discretization and Time-Dependent Problems | p. 266 |
| Introduction | p. 266 |
| Partial Discretization Applied to Boundary Value Problems | p. 267 |
| Time-Dependent Problems Via Partial Discretization | p. 270 |
| Analytical Solution Procedures | p. 276 |
| Finite Element Solution Procedures in the Time Domain | p. 283 |
| References | p. 307 |
| Suggested Further Reading | p. 308 |
| Generalized Finite Elements, Error Estimates, and Concluding Remarks | p. 309 |
| The Generalized Finite Element Method | p. 309 |
| The Discretization Error in a Numerical Solution | p. 310 |
| A Measure of Discretization Error | p. 311 |
| Estimate of Discretization Error | p. 313 |
| The State of the Art | p. 322 |
| References | p. 322 |
| Suggested Further Reading | p. 322 |
| Index | p. 323 |
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