Finite Element Method

Patrick Ciarlet is a professor at the Applied Mathematics Unit of ENSTA Paris, France. His research focuses on the design of numerical methods for solving problems in physics, modeled by systems of partial differential equations.Eric Lunéville is a professor at the Applied Mathematics Unit of ENSTA Paris, France. His research focuses on the design of numerical methods for solving problems in mechanics and physics, modeled by systems of partial differential equations.

• Preface ixChapter 1 Theoretical Aspects of Elliptic Equations 11.1. Some vocabulary 11.1.1. Boundary conditions for the Laplace equation 31.1.2. Other boundary conditions 41.2. Classical solutions to Laplace’s equation 51.2.1. Green's formulae 51.2.2. The maximum principle 81.3. Weak solutions 91.4. Reminders on Banach and Hilbert spaces 111.4.1. Banach spaces 121.4.2. Hilbert spaces 131.5. Sobolev spaces 161.5.1. Differentiation in the distribution sense 171.5.2. "Sufficiently regular" boundaries 231.5.3. Differentiation in the classical or distribution sense? 251.5.4. A compactness result 261.6. Existence of the trace and integration by parts formulae 271.7. Exercises – statements 35Chapter 2 Variational Formulations and Their Solutions 432.1. Variational formulations of the Dirichlet, Neumann and Fourier problems 432.1.1. Homogeneous Dirichlet problem 432.1.2. Non-homogeneous Dirichlet problem 462.1.3. Neumann and Fourier problems 472.2. Other variational formulations examples 502.2.1. Variable coefficient problems 502.2.2. Second-order elliptic problems 512.2.3. Higher order problems 512.2.4. Less-regular data 522.3. Existence and uniqueness of weak solutions 532.3.1. Neumann problem 542.3.2. Dirichlet problem – Poincaré’s inequality 562.3.3. The non-homogeneous Dirichlet problem 592.4. Existence and uniqueness – general framework 602.4.1. Lax–Milgram theorem 612.4.2. An example application of the Lax–Milgram theorem with the Laplacian 642.4.3. The Helmholtz problem 672.4.4. Well-posed problems 712.4.5. The Banach–Necas–Babuska theorem 722.4.6. Fredholm alternative 762.5. Some properties of weak solutions 782.5.1. Maximum principle 792.5.2. Regularity of the solutions 812.6. Exercises – statements 85Chapter 3 Introduction to the Finite Element Method 953.1. Galerkin’s approximation 953.1.1. Case of a coercive form 973.1.2. Case of a form satisfying the stability and solvability conditions 1013.1.3. Approximation example: Hilbert bases 1043.1.4. Different solution and test-function spaces 1053.2. Principles of the affine finite element method in two dimensions 1073.2.1. Mesh 1073.2.2. Global basis functions properties 1103.2.3. Norm of the elements of the approximation space 1123.2.4. Sparse matrix property 1133.2.5. Lagrange interpolation 1133.2.6. Local nature of the finite element method 1133.3. General procedure of constructing finite elements 1173.3.1 Lagrangian finite elements of order k 1173.3.2. Examples of finite elements 1183.3.3. Constructing the approximation space 1273.3.4. Two-dimensional examples of assembly 1293.4. Extension of finite elements 1333.4.1. Hermitian finite elements 1343.4.2. “Moment”-type finite elements 1353.4.3. Constructing the approximation space 1363.4.4. Vectorial finite elements 1373.5. Exercises – statements 137Chapter 4 Numerical Analysis of the Finite Element Method 1434.1. Convergence of finite element methods 1434.1.1. Local interpolation error 1454.1.2. Estimating errors in the finite element method 1524.1.3. “Piecewise regular” solutions 1564.2. Error estimators and mesh refinement 1604.3. Non-polyhedral domains and approximations of the data 1634.3.1. Non-polyhedral open subsets 1634.3.2. Isoparametric finite elements 1674.3.3. Approximation of the data 1684.4. Exercises – statements 172Chapter 5 Concrete Aspects of the Finite Element Method 1855.1. Implementation 1855.1.1. Finite element meshes 1865.1.2. Basic calculations 1895.1.3. Assembly of the global matrices and the right-hand side 1915.1.4. Eliminating the essential boundary conditions 1955.2. Algorithmic considerations 1995.2.1. A particular case 2005.2.2. The general 2D case 2005.2.3. The general 3D case 2055.3. Some numerical illustrations 2075.3.1. The Laplace equation 2085.3.2. The diffusion equation 2135.3.3. Bidimensional elasticity 2155.3.4. Some elementary tools for 2D meshes 222Appendices 231Appendix 1 Solving Linear Systems 233Appendix 2 Solutions 271Appendix 3 Formulas 369References 371Index 373