Finite Element Method and Applications

Introduction by Acad. Caius Iacob

Foreword

1. Functional Analysis Itinerary
   1.1. Vector spaces.
   1.2. Topological spaces. Topological vector spaces.
   1.3. Normed vector spaces. Banach and Hilbert spaces.
   1.4. Operators and imbeddings. Dual spaces.
   1.5. Continuous functions spaces.
   1.6. Lebesgue measure and integral in Rⁿ.
   1.7. Lp  spaces and generalized derivatives.
   1.8. Sobolev spaces, imbedding theorems, Friedrichs-Poincaré inequalities. Concept of trace and Green formulas.
   1.9. Distributions, fundamental solutions and Green functions.
   1.10. Linear variational problems. Lax-Milgram lemma.
   1.11. Elements of differential calculus in normed spaces.
   1.12. Elements of affine geometry.
2. The Finite Element Method (FEM) for Elliptic Equations
   2.1. Linear variational problems-abstract formulations.
   2.2. Variational formulation for second order problems.
   2.3. Variational formulation for fourth order problems.
   2.4. The bidimensional Stokes problem.
   2.5. The approximation of elliptic variational problems-general theory.
   2.6. The finite element method and its mathematical frame.
   2.7. Multidimensional Lagrange and Hermite interpolation.
   2.8. Interpolation polynomials (examples).
   2.9. The notion of finite element. Fundamental properties.
   2.10. The interpolation theory in Sobolev spaces.
   2.11. Second order problems in polygonal domains.
   2.12. Numerical integration in FEM.
   2.13. Isoparametric finite elements.
   2.14. Second order problems over curved domains.
   2.15. The approximation of fourth order problems.
3. Finite Element-Finite Differences Method (FDM) for Parabolic Equations
   3.1. The Cauchy-Dirichlet problem for heat equation-a variational formulation.
   3.2. Solving the variational problem.
   3.3. Error estimations and convergence for various discretization schemes.
   3.4. The efficiency of Crank-Nicolson and Calahan schemes.
4. Boundary Element Method (BEM)
   4.1. Introductory notions.
   4.2. Boundary element method for potential problems. Indirect formulation. Direct formulation. Two-dimensional problems. Linear elements. Quadratic and higher-order elements.
   4.3. Nonpotential problems treated by boundary element method. Diffusion equation. Elastostatics equations (Navier). Euler’s equations. Stokes model. Navier-Stokes equation.
   4.4. Coupling of BEM and FEM.
5. General Comments on FEM Implementation and Numerical Examples
   5.1. Strategies of FEM implementation
   5.2. Comparison between FEM and BEM
   5.3. Comparison between FEM, FDM and spectral methods (SM).
   5.4. Numerical examples.

References

Contents (in Russian)

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* IJNME – International Journal for N umerical Methods in Engineering
**INRIA – Institut National de Recherche en Informatique et en Automatique (France)
***CMAME  – Computer Methods in Applied Mechanics and Engineering