Page 11 - Basics of Fluid Mechanics and Introduction to Computational Fluid Dynamics

2.2 The Mathematical Foundation of the Least-Squares
Finite Element Method 363
2.3 Div-Curl (Rot) Systems 370
2.4 Div-Curl (Rot)-Grad System 375

2.5 Stokes’ Problem 377
3 Boundary Element Method (BEM) 380
3.1 Abstract Formulation of the Boundary Element
Method 381

3.2 Variant of the Complex Variables Boundary
Element Method [112] 385
3.3 The Motion of a Dirigible Balloon 389
3.4 Coupling of the Boundary Element Method and
the Finite Element Method 391

7. THE FINITE VOLUME METHOD AND
THE GENERALIZED DIFFERENCE METHOD 397

1 ENO Finite Volume Schemes 398
1.1 ENO Finite Volume Scheme in One Dimension 399

1.2 ENO Finite Volume Scheme in Multi-Dimensions 406
2 Generalized Difference Method 41]
2.1 Two-Point Boundary Value Problems 411

2.2 Second Order Elliptic Problems 424
2.3 Parabolic Equations 429
2.4 Application 433

8 SPECTRAL METHODS 439
1 Fourier Series 442

1.1 The Discretization 442
1.2 Approximation of the Derivatives 445
2 Orthogonal Polynomials 447

2.1 Discrete Polynomial Transforms 447
2.2 Legendre Polynomials 450
2.3 Chebyshev Polynomials 452

3 Spectral Methods for PDE 455
3.1 Fourier—Galerkin Method 455
3.2 Fourier-Collocation 456

3.3 Chebyshev-Tau Method 457
3.4 Chebyshev-Collocation Method 458
3.5 The Calculation of the Convolution Sums 459

3.6 Complete Discretization 460