Читать книгу An Introduction to the Finite Element Method for Differential Equations - Mohammad Asadzadeh - Страница 4

1 Chapter 1Figure 1.1 Tricomi equation: an example of a variable coefficient classifica...Figure 1.2 Outward unit normal at a point .Figure 1.3 A heat‐conducting one‐dimensional wire.Figure 1.4 A vibrating string.

2 Chapter 2Figure 2.1 The hat function over the interval .Figure 2.2 Illustrating the existence of a unique solution for (V) and (M)....

3 Chapter 3Figure 3.1 Linear Lagrange basis functions for .Figure 3.2 The linear interpolant on a single interval.Figure 3.3 An example of a function in .Figure 3.4 A general piecewise linear basis function .Figure 3.5 A partition of .Figure 3.6 Piecewise linear basis functions.Figure 3.7 and Figure 3.8 and .Figure 3.9 (a) Linear interpolation and (b) basis functions for .Figure 3.10 Linear Lagrange basis functions for .Figure 3.11 Piecewise linear interpolant of .Figure 3.12 Linear Lagrange basis functions for on subinterval .Figure 3.13 Example of a projection onto .Figure 3.14 An example of a function and its projection in .Figure 3.15 Midpoint approximation of the integral .Figure 3.16 Trapezoidal approximation of the integral .Figure 3.17 Simpson's rule approximation of the integral .Figure 3.18 Identification of subintervals for composite Simpson's rule.Figure 3.19 Coefficients for composite Simpson's rule.

4 Chapter 5Figure 5.1 A partition of into nonuniform subintervals.Figure 5.2 The basis functions .Figure 5.3 , and .Figure 5.4 The basis test and trial functions for this problem.

5 Chapter 6Figure 6.1 Example of finite difference schemes for population model. E.E., ...Figure 6.2 Two continuous linear Galerkin approximations of .Figure 6.3 The jump and the right and left limits .Figure 6.4 Time directions in forward and dual solutions.Figure 6.5 Orthogonality: .

6 Chapter 7Figure 7.1 A decreasing temperature profile with data .Figure 7.2 and its linear interpolant (both in and ) .Figure 7.3 Quadratic Lagrange basis functions .Figure 7.4 The linear time (a) and space (b) basis functions.Figure 7.5 Flow of cars with on highway.Figure 7.6 Characteristic line.Figure 7.7 A boundary layer.Figure 7.8 Upstreams transported oscillatory approximate solution.Figure 7.9 A ‐telted mesh in a slab .

7 Chapter 8Figure 8.1 A smooth domain with an outward unit normal .Figure 8.2 A rectangular domain with outward unit normals to its sides.Figure 8.3 A linear function on a subinterval Figure 8.4 Example of triangulation of a domain.Figure 8.5 A triangle in as a piecewise linear function and its projection...Figure 8.6 A linear basis function in a triangulation in .Figure 8.7 Uniform triangulation of with .Figure 8.8 Support of a single basis function in Example 8.1.Figure 8.9 (a) Uniform triangulation of and (b) the reference element .Figure 8.10 The nodal interpolant of in case.

8 Chapter 9Figure 9.1 The orthogonal ( ) projection of on Figure 9.2 The nodal interpolant of in case.Figure 9.3 The adaptivity principle: to refine mesh for large .Figure 9.4 Rectangular domain with outward unit normal to its sides.Figure 9.5 A uniform triangulation of .Figure 9.6 The considered triangulation for .Figure 9.7 A uniform mesh of square (a) and its standard element (b).

9 2Figure B.1 Standard basis functions and .