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| Continuum Boundary Value Problems and the Need for Numerical Discretization. Finite Difference Methods | |
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| Introduction | |
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| Some Examples of Continuum Problems | |
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| Finite Differences in One Dimension | |
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| Derivative Boundary Conditions | |
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| Nonlinear Problems | |
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| Finite Differences in More Than One Dimension | |
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| Problems Involving Irregularly Shaped Regions | |
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| Nonlinear Problems in More Than One Dimension | |
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| Approximation and Convergence | |
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| Concluding Remarks | |
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| References | |
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| Suggested Further Reading | |
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| Weighted Residual Methods: Use of Continuous Trial Functions | |
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| Introduction-Approximation by Trial Functions | |
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| Weighted Residual Approximations | |
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| Approximation to the Solutions of Differential Equations and the Use of Trial Function-Weighted Residual Forms. Boundary Conditions Satisfied by Choice of Trial Functions | |
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| Simultaneous Approximation to the Solutions of Differential Equations and to the Boundary Conditions | |
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| Natural Boundary Conditions | |
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| Boundary Solution Methods | |
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| Systems of Differential Equations | |
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| Nonlinear Problems | |
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| Concluding Remarks | |
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| References | |
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| Suggested Further Reading | |
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| Piecewise Defined Trial Functions and the Finite Element Method | |
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| Introduction-The Finite Element Concept | |
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| Some Typical Locally Defined Narrow-Base Shape Functions | |
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| Approximation to Solutions of Differential Equations and Continuity Requirements | |
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| Weak Formulation and the Galerkin Method | |
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| Some One-Dimensional Problems | |
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| Standard Discrete System. A Physical Analogue of the Equation Assembly Process | |
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| Generalization of the Finite Element Concepts for Two- and Three-Dimensional Problems | |
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| The Finite Element Method for Two-Dimensional Heat Conduction Problems | |
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| Two-Dimensional Elastic Stress Analysis Using Triangular Elements | |
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| Are Finite Differences a Special Case of the Finite Element Method? | |
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| Concluding Remarks | |
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| References | |
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| Suggested Further Reading | |
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| Higher Order Finite Element Approximation | |
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| Introduction | |
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| Degree of Polynomial in Trial Functions and Convergence Rates | |
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| The Patch Test | |
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| Standard Higher Order Shape Functions for One-Dimensional Elements with C[superscript 0] Continuity | |
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| Hierarchical Forms of Higher Order One-Dimensional Elements with C[superscript 0] Continuity | |
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| Two-Dimensional Rectangular Finite Element Shape Functions of Higher Order | |
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| Two-Dimensional Shape Functions for Triangles | |
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| Three-Dimensional Shape Functions | |
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| Concluding Remarks | |
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| References | |
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| Suggested Further Reading | |
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| Mapping and Numerical Integration | |
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| The Concept of Mapping | |
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| Numerical Integration | |
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| More on Mapping | |
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| Mesh Generation and Concluding Remarks | |
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| References | |
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| Suggested Further Reading | |
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| Variational Methods | |
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| Introduction | |
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| Variational Principles | |
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| The Establishment of Natural Variational Principles | |
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| Approximate Solution of Differential Equations by the Rayleigh-Ritz Method | |
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| The Use of Lagrange Multipliers | |
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| General Variational Principles | |
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| Penalty Functions | |
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| Least-Squares Method | |
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| Concluding Remarks | |
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| References | |
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| Suggested Further Reading | |
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| Partial Discretization and Time-Dependent Problems | |
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| Introduction | |
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| Partial Discretization Applied to Boundary Value Problems | |
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| Time-Dependent Problems Via Partial Discretization | |
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| Analytical Solution Procedures | |
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| Finite Element Solution Procedures in the Time Domain | |
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| References | |
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| Suggested Further Reading | |
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| Generalized Finite Elements, Error Estimates, and Concluding Remarks | |
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| The Generalized Finite Element Method | |
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| The Discretization Error in a Numerical Solution | |
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| A Measure of Discretization Error | |
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| Estimate of Discretization Error | |
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| The State of the Art | |
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| References | |
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| Suggested Further Reading | |
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| Index | |