The finite element method and its reliability Ivo Babuéska and Theofanis Strouboulis

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Machine generated contents note: 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 -- 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 -- .epilogue -- / Appendix to chapter 2 -- Appendix to chapter 3 -- Glossary of symbols -- Author Index -- Subject Index