[FreeFEM 5] Implementing weak formulation to solve partial differential equations in FreeFEM
Ғылым және технология
After discussing mesh generation and finite element spaces, everything is ready to jump into the real implementation of solving a partial differential equation (PDE) using the finite element method. In this video, we have a look at simply FreeFEM can be used to implement the weak (variational) form of PDEs in order to solve them numerically. Let’s go for it.
Codes, models, and resources:
You can find all the codes and models required to follow the videos and reproduce the output, grouped for different episodes, at tuxriders.com/videos/freefem/ and github.com/TuxRiders/freefem-...
💡 Finite element series, in which a single problem was solved in different open-source tools including FreeFEM • Introduction to finite...
💡 Applied numerical computing series, in which the underlying theories of these videos are discussed: • Introduction to applie...
Topics covered:
🎯 Writing the variational form of PDEs in FreeFEM
🎯 Function approximation using the weak formulation
🎯 Various ways to solve a PDE in FreeFEM
🎯 Solving a steady diffusion problem in FreeFEM
🎯 Solving PDEs on a surface mesh
🎯 Increasing the accuracy of the solution
Lecturer: Mojtaba Barzegari mbarzegary.github.io/
To learn more about the goals of the TuxRiders project, please visit our website at tuxriders.com.
Chapters in this video!
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00:00 - Intro
01:08 - Function approximation using the weak formulation
07:00 - Solving steady diffusion PDE using the solve keyword
12:43 - Solving the PDE using the problem keyword
14:03 - Solving the PDE using the varf keyword
18:37 - Solving a PDE on a surface mesh
![[FreeFEM 6] Adaptive mesh generation in FreeFEM](/img/d.gif)
Пікірлер: 13
when the function f is depending on the solution u , how we do it
@TuxRiders
Жыл бұрын
then it becomes part of the bilinear term and moves to left hand side part. such formulation is common in reaction-diffusion problems (for the reaction terms).
So, are Neumann boundary conditions always taken into account through the weak formulation of the problem? Great videos by the way!
@TuxRiders
Жыл бұрын
yes, you can embed your Neumann BC in the weak form of the PDE you are solving.
Hi, I had been search how solve the Wave Equation 2D, with FreeFem++ but I didn't find nothing, could you explain it please, I would be grateful to you
@TuxRiders
Жыл бұрын
yes, sure. I will start to talk about some minimal FreeFEM projects in the near future.
how to solve a system of coupled equations , for eg navier stokes eqaution
@TuxRiders
6 ай бұрын
I will create some videos on coupled problems soon. there are various ways to do so, but in the simplest form, you can solve the equations one by one while they have connected state variables, like diffusion convection models where the velocity comes from navier stokes.
If there is no source term (e.g. if f=0), would you just write X[ ] =A^-1 and not worry about defining vector b?
@JA-bc5oc
11 ай бұрын
Or do you still define b as shown and freeFEM will just set it to zero? One further question, how do you deal with a system PDEs in multiple unknowns using the varf keyword? How would you define the matrix A, the vector b and the unknown X in this case? When using the 'problem' keyword, you combine all equations into a single integral, but I'm not sure if you'd do the same when using the varf keyword.
@TuxRiders
10 ай бұрын
so, for the first question: you need to have `b` anyway even if `f` is zero. it contains other important info such as boundary conditions, without which the problem is not well-imposed.
@TuxRiders
10 ай бұрын
for the second question (system of PDEs): yes, you can either write them in one `varf` statement or solve them one by one. we will have videos on this soon.
@JA-bc5oc
10 ай бұрын
@@TuxRiders Thanks for your reply! Great, a video on this would be awesome.