微积分网课代修|偏微分方程代写Partial Differential Equation代考|MATH480 Finite Element Method (FEM) Basics

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微积分网课代修|偏微分方程代写Partial Differential Equation代考|Finite Element Method (FEM) Basics

The core Partial Differential Equation Toolbox algorithm is a PDE solver that uses the Finite Element Method (FEM) for problems defined on bounded domains in the plane.
The solutions of simple PDEs on complicated geometries can rarely be expressed in terms of elementary functions. You are confronted with two problems: First you need to describe a complicated geometry and generate a mesh on it. Then you need to discretize your PDE on the mesh and build an equation for the discrete approximation of the solution. The PDE app provides you with easy-to-use graphical tools to describe complicated domains and generate triangular meshes. It also discretizes PDEs, finds discrete solutions and plots results. You can access the mesh structures and the discretization functions directly from the command line (or from a file) and incorporate them into specialized applications.
Here is an overview of the Finite Element Method (FEM). The purpose of this presentation is to get you acquainted with the elementary FEM notions. Here you find the precise equations that are solved and the nature of the discrete solution. Different extensions of the basic equation implemented in Partial Differential Equation Toolbox software are presented. A more detailed description can be found in “Elliptic Equations” on page 5-2, with variants for specific types in “Systems of PDEs” on page 5-13, “Parabolic Equations” on page 5-17, “Hyperbolic Equations” on page 5-20, “Eigenvalue Equations” on page 5-22, and “Nonlinear Equations” on page 5-26.
You start by approximating the computational domain $\Omega$ with a union of simple geometric objects, in this case triangles (2-D geometry) or tetrahedra (3-D geometry). (This discussion applies to both triangles and tetrahedra, but speaks of triangles.) The triangles form a mesh and each vertex is called a node. You are in the situation of an architect designing a dome. The architect has to strike a balance between the ideal rounded forms of the original sketch and the limitations of the simple building-blocks, triangles or quadrilaterals. If the result does not look close enough to a perfect dome, the architect can always improve the result by using smaller blocks.
Next you say that your solution should be simple on each triangle. Polynomials are a good choice: they are easy to evaluate and have good approximation properties on small domains. You can ask that the solutions in neighboring triangles connect to each other continuously across the edges. You can still decide how complicated the polynomials can be. Just like an architect, you want them as simple as possible. Constants are the simplest choice but you cannot match values on neighboring triangles. Linear functions come next. This is like using flat tiles to build a waterproof dome, which is perfectly possible.

Now you use the basic elliptic equation (expressed in $\Omega$ )
$$-\nabla \cdot(c \nabla u)+a u=f .$$
If $u_{h}$ is the piecewise linear approximation to $u$, it is not clear what the second derivative term means. Inside each triangle, $\nabla u_{h}$ is a constant (because $u_{h}$ is flat) and thus the second-order term vanishes. At the edges of the triangles, $c \nabla u_{h}$ is in general discontinuous and a further derivative makes no sense.

微积分网课代修|偏微分方程代写Partial Differential Equation代考|Open the PDE App

For basic information on 2-D geometry construction, see “Create 2-D Geometry” on page $2-17$

Partial Differential Equation Toolbox software includes the PDE app, which covers all aspects of the PDE solution process. You start it by typing
pdetool
at the MATLAB command line. It may take a while the first time you launch the PDE app during a MATLAB session. The following figure shows the PDE app as it looks when you start it.

At the top, the PDE app has a pull-down menu bar that you use to control the modeling. Below the menu bar, a toolbar with icon buttons provide quick and easy access to some of the most important functions.

To the right of the toolbar is a pop-up menu that indicates the current application mode. You can also use it to change the application mode. The upper right part of the PDE app also provides the $x$ – and $y$-coordinates of the current cursor position. This position is updated when you move the cursor inside the main axes area in the middle of the PDE app.
The edit box for the set formula contains the active set formula.
In the main axes you draw the 2-D geometry, display the mesh, and plot the solution.
At the bottom of the PDE app, an information line provides information about the current activity. It can also display help information about the toolbar buttons.

微积分网课代修|偏微分方程代写Partial Differential Equation代 考|Finite Element Method (FEM) Basics

$$-\nabla \cdot(c \nabla u)+a u=f .$$

pdetool来启动它。