## The Finite Element Method: Theory, Implementation, and ApplicationsThis 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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### Contents

1 | |

Chapter
2 The Finite Element Method in 1D | 23 |

Chapter
3 Piecewise Polynomial Approximation in 2D | 45 |

Chapter
4 The Finite Element Method in 2D | 70 |

Chapter
5 TimeDependent Problems | 113 |

Chapter
6 Solving Large Sparse Linear Systems | 143 |

Chapter
7 Abstract Finite Element Analysis | 177 |

Chapter
8 The Finite Element | 202 |

Chapter
10 Transport Problems | 240 |

Chapter
11 Solid Mechanics | 257 |

Chapter
12 Fluid Mechanics | 289 |

Chapter
13 Electromagnetics | 326 |

Chapter
14 Discontinuous Galerkin Methods | 355 |

Appendix
A Some Additional Matlab Code | 373 |

378 | |

382 | |

### Other editions - View all

The Finite Element Method: Theory, Implementation, and Applications Mats G. Larson,Fredrik Bengzon No preview available - 2013 |

The Finite Element Method: Theory, Implementation, and Applications Mats G. Larson,Fredrik Bengzon No preview available - 2012 |

### Common terms and phrases

assume bilinear boundary conditions Cauchy-Schwarz inequality Cholesky factorization coefﬁcients coercivity constant continuous piecewise linear convergence deﬁned deﬁnition derive differential equations Dirichlet Dirichlet boundary conditions discrete domain edge efﬁciently eigenvalues element solution uh entries Exercise ﬁeld ﬁll-in ﬁnd ﬁnite element approximation Finite Element Method ﬁnite element solution ﬁrst ﬂow ﬂuid formula G Vh Galerkin method Galerkin orthogonality given gradient Heat equation Helmholtz decomposition Hilbert space Implementation inf-sup condition integrating interpolation error iteration KG‘K Lagrange Lax-Milgram lemma linear form linear system load vector LZ-projection mass matrix MATLAB mesh reﬁnement Nédélec Newton’s method nodal values non-linear norm null space number of nodes obtain parameter Ph f piecewise linear functions posteriori error estimate preconditioner pressure priori error estimate problem quadrature reﬁned routine satisﬁes the estimate shape functions so-called solving stiffness matrix subspace Theorem triangle unique variational formulation velocity Vv G V weak form zero