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First of all, FreeFem is a numerical computing software which allows a fast and automatized treatment of a variety of problems related to partial differential equations. Its name, FreeFem, speaks for itself: it is free and it uses the finite element method. Here are a few reasons for which you may choose to use FreeFem for a certain task:.
Before showing a first example, you need to install FreeFem. If you are not familiar with command line work or you just want to get to work, like me, you can install the visual version of FreeFem which is available here.
Of course, you can find example programs in the FreeFem manual or by making a brief search on the internet. Once we have the structure of the program, it is possible to change the shape of the domain in no time.
Define the geometry. It is possible to define the geometry of the domain in multiple ways. Then we simply write. This may help us later on when we define the problem or when we want to compute an integral over this specific part of the boundary. It is important to note that using labels we can impose different boundary conditions on several parts of the domain.
You can identify the command buildmesh and the parameter dictates the number of uniform discretization points taken on the curve C. Once the mesh is built you can visualize it using the command plot Th. Note that in FreeFem, like in C, every variable which appears for the first time must be specified by its type.
Now we have the geometry and the mesh, so we can write the problem. First we define a finite element space on the constructed triangulation by using the command. Note that P1 simply means we use P1 finite elements. Once we have a finite element space we can define variables in this space. Since we want to be able to impose a general boundary condition we define a function by using the command.
Note that whenever the variables x and y are used they represent the usual coordinates in the plane, so you cannot use them for something else. All the basic functions like exp, sin, cos are already implemented in FreeFem so you can use them at will.
We are ready to state the problem in its variational form. Since we want to solve with the boundary condition , we consider the weak form of the problem and we introduce this weak form in FreeFem using the commands.
We note the use of the keyword problem to denote a weak form of an EDP. There are several other things present in the above command. Note that to integrate over a two dimensional mesh we simply use the int2d command with the first argument the mesh and the second argument the function we wish to integrate on that mesh.
Note that we used variables defined on the finite element space on the corresponding mesh. Here are some simple tricks which allow to compute numerically the area and the perimeter of the triangulation. The idea is to use a dummy variable which is equal to 1 everywhere and then integrate it on the domain to find the area and on the boundary to find the perimeter.
The on operator allows us to put a boundary condition on parts of the boundary specified by the corresponding label. In order to solve the stated problem simply type its name:.
Visualize the results. Now the variable uh contains the numerical solution of our problem. In order to visualize it just type. This will produce a 2D plot with colors corresponding to the values of the function uh. In order to solve different problems, it suffices to change the problem formulation or the boundary condition.
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The fruit of a long maturing process, freefem , in its last avatar, FreeFEM , is a high level integrated development environment IDE for numerically solving partial differential equations PDE in dimension 1,2 3 and surface and line 3D. It is the ideal tool for teaching the finite element method but it is also perfect for research to quickly test new ideas or multi-physics and complex applications. Hyperbolic and parabolic problems are solved by iterative algorithms prescribed by the user with the high level language of FreeFEM. It has several triangular finite elements, including discontinuous elements. Everything is there in FreeFEM to prepare research quality reports with online color display, zooming and other features as well as postscript printouts. This manual is meant for students at a Masters level, for researchers at any level, and for engineers including financial engineering with some understanding of variational methods for partial differential equations.