Ebook: The Finite Element Method: Theory, Implementation, and Applications

This book gives an introduction to the finite element method as a general computational method for solving partial differential equations approximately. Our approach is mathematical in nature with a strong focus on the underlying mathematical principles, such as approximation properties of piecewise polynomial spaces, and variational formulations of partial differential equations, but with a minimum level of advanced mathematical machinery from functional analysis and partial differential equations. In principle, the material should be accessible to students with only knowledge of calculus of several variables, basic partial differential equations, and linear algebra, as the necessary concepts from more advanced analysis are introduced when needed. Throughout the text we emphasize implementation of the involved algorithms, and have therefore mixed mathematical theory with concrete computer code using the numerical software MATLAB is and its PDE-Toolbox. We have also had the ambition to cover some of the most important applications of finite elements and the basic finite element methods developed for those applications, including diffusion and transport phenomena, solid and fluid mechanics, and also electromagnetics.​

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This book gives an introduction to the finite element method as a general computational method for solving partial differential equations approximately. Our approach is mathematical in nature with a strong focus on the underlying mathematical principles, such as approximation properties of piecewise polynomial spaces, and variational formulations of partial differential equations, but with a minimum level of advanced mathematical machinery from functional analysis and partial differential equations. In principle, the material should be accessible to students with only knowledge of calculus of several variables, basic partial differential equations, and linear algebra, as the necessary concepts from more advanced analysis are introduced when needed. Throughout the text we emphasize implementation of the involved algorithms, and have therefore mixed mathematical theory with concrete computer code using the numerical software MATLAB is and its PDE-Toolbox. We have also had the ambition to cover some of the most important applications of finite elements and the basic finite element methods developed for those applications, including diffusion and transport phenomena, solid and fluid mechanics, and also electromagnetics.?

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This book gives an introduction to the finite element method as a general computational method for solving partial differential equations approximately. Our approach is mathematical in nature with a strong focus on the underlying mathematical principles, such as approximation properties of piecewise polynomial spaces, and variational formulations of partial differential equations, but with a minimum level of advanced mathematical machinery from functional analysis and partial differential equations. In principle, the material should be accessible to students with only knowledge of calculus of several variables, basic partial differential equations, and linear algebra, as the necessary concepts from more advanced analysis are introduced when needed. Throughout the text we emphasize implementation of the involved algorithms, and have therefore mixed mathematical theory with concrete computer code using the numerical software MATLAB is and its PDE-Toolbox. We have also had the ambition to cover some of the most important applications of finite elements and the basic finite element methods developed for those applications, including diffusion and transport phenomena, solid and fluid mechanics, and also electromagnetics.?
Content:
Front Matter....Pages i-xvii
Piecewise Polynomial Approximation in 1D....Pages 1-22
The Finite Element Method in 1D....Pages 23-44
Piecewise Polynomial Approximation in 2D....Pages 45-69
The Finite Element Method in 2D....Pages 71-111
Time-Dependent Problems....Pages 113-142
Solving Large Sparse Linear Systems....Pages 143-176
Abstract Finite Element Analysis....Pages 177-201
The Finite Element....Pages 203-223
Non-linear Problems....Pages 225-239
Transport Problems....Pages 241-256
Solid Mechanics....Pages 257-287
Fluid Mechanics....Pages 289-325
Electromagnetics....Pages 327-354
Discontinuous Galerkin Methods....Pages 355-372
Back Matter....Pages 373-385

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This book gives an introduction to the finite element method as a general computational method for solving partial differential equations approximately. Our approach is mathematical in nature with a strong focus on the underlying mathematical principles, such as approximation properties of piecewise polynomial spaces, and variational formulations of partial differential equations, but with a minimum level of advanced mathematical machinery from functional analysis and partial differential equations. In principle, the material should be accessible to students with only knowledge of calculus of several variables, basic partial differential equations, and linear algebra, as the necessary concepts from more advanced analysis are introduced when needed. Throughout the text we emphasize implementation of the involved algorithms, and have therefore mixed mathematical theory with concrete computer code using the numerical software MATLAB is and its PDE-Toolbox. We have also had the ambition to cover some of the most important applications of finite elements and the basic finite element methods developed for those applications, including diffusion and transport phenomena, solid and fluid mechanics, and also electromagnetics.?
Content:
Front Matter....Pages i-xvii
Piecewise Polynomial Approximation in 1D....Pages 1-22
The Finite Element Method in 1D....Pages 23-44
Piecewise Polynomial Approximation in 2D....Pages 45-69
The Finite Element Method in 2D....Pages 71-111
Time-Dependent Problems....Pages 113-142
Solving Large Sparse Linear Systems....Pages 143-176
Abstract Finite Element Analysis....Pages 177-201
The Finite Element....Pages 203-223
Non-linear Problems....Pages 225-239
Transport Problems....Pages 241-256
Solid Mechanics....Pages 257-287
Fluid Mechanics....Pages 289-325
Electromagnetics....Pages 327-354
Discontinuous Galerkin Methods....Pages 355-372
Back Matter....Pages 373-385
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