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1 Introduction 1.0 Introduction 1.1 A brief history of the finite element method 1.2 On computational engineering 2 Mathematical formulation of the model problem 2.0 Introduction 2.1 The one-dimensional model problem 2.1* The two-dimensional model problem 2.2 The variational formulation of the model problem 2.2* The variational formulation of the model problem in two dimensions 2.3 Smoothness of the solution of the model problem 2.3* Smoothness of the solution of the two-dimensional model problem 2.4 Historical and bibliographical remarks 3 The finite element method 3.0 Introduction 3.1 The Galerkin method and the properties of the approximate solution 3.1* The Galerkin method in two dimensions 3.2 The finite element method 3.2* The finite element method in two dimensions 3.3 The finite element interpolation and approximation error 3.3* The finite element interpolation and approximation error in two dimensions 3.4 The finite element interpolation and approximation: a special case 3.4* The interpolation error for the function ragp(0) in two dimensions 3.5 The description of the mesh and analyses of its optimality 3.5* The description of the mesh and analysis of its optimality in two dimensions 3.6 Historical and bibliographical comments 4 Local behavior in the finite element method 4.0 Introduction 4.1 A-priori estimates for the nodal errors and the maximum error 4.1* A-priori estimates for the nodal errors and the maximum error in two dimensions 4.2 A-priori estimates for negative norms of the error 4.2* A-priori estimates for negative norms of the error in two dimensions 4.3 Local and pollution error in the finite element solution 4.3* The splitting of the error in the two-dimensional case 4.4 A-priori estimates for the pollution error 4.4* A-priori estimates for the pollution error in two dimensions 4.5 Further analysis of the pollution error: Interior estimates ·· r 4.5* The interior estimates in two dimensions 4.6 A-priori estimates for the local error: superconvergence 4.6* Superconvergence in two dimensions 4.7 A general approach for the analysis of local behavior of finite element methods for locally periodic meshes and its applications 4.7* A general approach for local asymptotic analysis and analysis of the superconvergence in two dimensions 4.8 Superconvergence via local averaging 4.8* Superconvergence based on local recovery: the two-dimensional case 4.9 Historical and bibliographical comments 5 A-posteriori estimation of the error 5.0 Introduction 5.1 The residuum and the construction of lower and upper estimators for the energy norm of the error 5.1* The residuum and the construction of lower and upper estimators for the energy norm of the error: the two dimensional case 5.2 Analysis of the Dirichlet element residual estimator 5.2* Analysis of the Dirichlet element residual estimator in two dimensions 5.3 Analysis of the Neumann element residual estimator 5.3* Analysis of the Neumann element residual estimator in two dimensions 5.4 Subdomain residual estimates 5.4* Subdomain residual estimates in two dimensions 5.5 Analysis of the explicit element residual estimator 5.5* Analysis of the explicit element residual estimator in two dimensions 5.6 A-posteriori error estimates based on local averaging 5.6* A-posteriori estimates based on local averaging: the two-dimensional case 5.7 The effectivity of the error indicators: the principles for their comparison and the detailed analysis of their quality 5.7* The asymptotic analysis of the effectiveness of the element error indicators in two dimensions 5.8 Historical and bibliographical comments 6 Guaranteed a-posteriori error estimation and a-posteriori estimation of the pollution error 6.0 Introduction 6.1 Upper and lower bounds for the energy norm of the error 6.1* Upper and lower bounds for the energy norm of the error in two dimensions 6.2 Upper and lower bounds for the error in the output, and a-posteriori estimation of the pollution 6.2* Upper and lower bounds for the error in the outputs, and a-posteriori estimation of the pollution error in two dimensions 6.3* Historical and bibliographical comments 1: \ " * - 1: * : , I f 1' . ir: 1 ' .. i l |. | :' |1 ' ' l r ! !'^ .epilogue / Appendix to chapter 2 Appendix to chapter 3 Glossary of symbols Author Index Subject Index