Відео

It depends on which PDE you want to solve. For the Poisson equation there is plenty of stuff on GitHub. If you want something that is more versatile, I would go for FEniCS/FEniCSx, Ngsolve, Dune, MFEM or deal.II, but some of the latter might not have Python bindings yet. Personally, I have mostly worked with FEniCS in Python and deal.II in C++. If you can implement your PDE in FEniCS on your first try, then FEniCS is my favorite, but sometimes I needed low-level access to the library and found deal.II to be more convenient. Under the hood both libraries use C++ anyway.

@@numericalanalysisbyjulianr4242 perfect, i will look into these. Thank you so much for the awesome video and your quick reply !

@@numericalanalysisbyjulianr4242 i thought I had replied to you but I can't see my comment when checking again : just wanted to thank you one more time for the helpful answer, you rock !

Do you mean that the matrix A should be transposed? If this is what you meant, it does not make any difference here since the bilinear form is symmetric and the linear system is also symmetric. In general, one needs to be cautious though with the assembly of the linear system, since for more complicated problems the order of i and j matters. That's why if you derive the weak form by multiplying from the right with a test function, like in this video, you need to switch i and j in your code in the assembly of the linear system. Thus, many engineers multiply with a test function from the left, i.e. (φ, Δu), to later avoid any confusions in the assembly of the linear system.

@@numericalanalysisbyjulianr4242 Thank you so much!

That depends on the problem that you are solving. For example, if you have a uniform gravitational force pushing down you could have something like f(x, y) = -1 and thus f(-0.75, -0.75) = -1.

Thank you so much got it ❤

Yes, if you solve the weak form of a differential equation with the same trial and test space, i.e. the same function space for u and φ, this is called Galerkin method. Some people also use different function spaces for u and φ, this is then called Petrov-Galerkin method.

Yes, the minus sign was missing there in the video.

For time-dependent problems you can take a look at chapter 12 and especially section 12.3.2 in the lecture notes doi.org/10.15488/11709

Almost correct. The weak form is defined in H^1(Ω) resp. H^1_0(Ω) when we include the boundary conditions. As you correctly observed, for the weak form we only need to use first derivatives of the solution. But since these derivatives only need to make sense when we integrate over Ω, e.g. (u', Φ') needs to make sense, it is sufficient if the solution u is once *weakly* differentiable. We denote this then by H^1 for the space of function that are once weakly differentiable. If you want more details on weak derivatives and the space H^1, you can take a look at Sections 7.1.4 and 7.1.5 in the lecture notes from the video description.

Thank you a lot. H¹ is Sobolev space, right? I ask you this because I use another letter in my book

@@nick45be Yes, H^1 is a Sobolev space. Sometimes people also write Sobolev spaces as W^{k,p} with k denoting how often it is differentiable and p denoting that these derivatives need to be in L^p. For many PDEs, we just need p = 2 and hence the short hand notation H^k = W^{k,2} is used.

@@numericalanalysisbyjulianr4242 🤩🤩🤩

Yes, this is correct. For the finite element method we cannot directly work with the strong form in C^2(Ω) \cap C^1(\bar{Ω}) but instead use the weak form and solve it in H^1_0(Ω). (Note: There are also methods that work directly with the strong form, e.g. the Finite Difference method.) So far we have not used the energy minimzation anywhere and could ignore that. Nevertheless, especially in engineering, many people start off with some functional that needs to be minimized and derive the weak form from this energy minimization. Furthermore, sometimes one needs to use the energy minimization to prove theoretical results, e.g. the existence of solutions.

@@numericalanalysisbyjulianr4242 thank you very much your answer and video are awesome!

You can check out my online explanation of multigrid for the Finite Element Method at julianroth.org/documentation/multigrid/index.html. The code for this is also on GitHub

You are almost right. Solutions of the strong formulation need to be twice continuously dofferentiable and solutions of the weak formulation need to be once weakly differentiable. So automatically each strong solution is also a weak solution. But a weak solution is only a strong solution if it is also twice continuously differentiable. Otherwise you can't perform integration by parts and recover the Δu term. TL;DR: For u in C^2(Ω) and C^1(\bar{Ω}) the strong and weak form are equivalent.

Yes, your summary is correct. With FDM we just approximate the derivatives in the strong form of the PDE and with FEM we work with element-wise polynomial approximations. We don't construct an eigenvalue problem though but a linear equation system. A big difference is that FDM uses the strong formulation of the PDE, e.g. -u" = f, whereas FEM uses the weak/variational formulation of the PDE, e.g. (u', φ') = (f, φ). Like you mentioned, imho FEM is more flexible when it comes to meshes, since FDM requires "structured" grids. With FEM it is easier to deal with more complicated geometries, e.g. plate with hole. This has then also many engineering applications, e.g. Isogemetric Analysis (IGA) uses spline basis functions to work on CAD generated domains directly.

Yes, you are correct the RHS should be integrated over the master element K(1). This is a mistake in the video but at least it is later correct for the Laplace/stiffness term. Thank you for pointing this out 🤗

For future references, please provide concrete examples to make answering your question easier. Nevertheless, here is a very general answer: The choice of the finite element basis depends on your function space, which in turn depends on your PDE. At the example of the Poisson equation, here we have a second order PDE, but for the weak formulation we apply partial integration and thus have at most only first derivatives. This needs to be reflected by the function space and we choose the space H^1 (or H^1_0 if we have homogeneous Dirichlet BC) which contains all functions whose first (weak) derivative is L^2 integrable. We can thus choose the Lagrange FE basis of piecewise polynomials of up to order r which is a conforming subspace of H^1, i.e. all functions from our FE space are also contained in H^1. If we have a fourth order PDE like the biharmonic problem, we apply integration by parts and arrive at a weak formulation which has up to second order derivatives. By the same argumentation as above, we thus use H^2 as our function space, since the second order derivatives need to be L^2-integrable. Now we can't use the same FE basis anymore as for the Poisson problem, since between elements we only ensure continuity of the FE basis (C^0), but we would need also the continuity of the derivatives of the FE basis (C^1). Therefore you need to use C^1 conforming FE, like the Hermite basis, or split the problem into a mixed system similar to fenicsproject.org/olddocs/dolfin/1.6.0/python/demo/documented/biharmonic/python/documentation.html. The problem of choosing an appropriate more complicated finite element is not only specific to fourth order PDEs, but also applies to other problems like Maxwell's equations from electromagnetism, where for the simplest problem (Eddy-Current) one solves - curl(curl u) = f and after integration by parts we need a function space where curl u is L^2-integrable (H^curl). There you can't use the standard Langrange basis either, but need Nedelec elements to be H^curl conforming.

@@numericalanalysisbyjulianr4242 thank you so much... I used biharmonic equation. But i cant create a stiffness matrix. Because i don't know the basis when i used discrete finite element method

My example is in hilbert space of fourth order The problem is biharmonic plus poisson =f I used green theorem to obtain weak form twice for biharmonic and once for poission Then i have galerkin equation B(y, v) =F(v) Vn is a subspace(??? ) of H2 Yn is a basis of linear combination What is a basis of finite element method? To creat stiffness matrix Is it yn=sum Cj Yj, and j =1,..., n??

@@emanalquraishi9279 Yes, in your case your FE space needs to be a subspace of H^2. For this you could use the cubic Hermite basis: hplgit.github.io/INF5620/doc/pub/H14/fem/html/._main_fem007.html. If you don't want to use C^1 FE, then you could also transform the bihramonic into two second order equations and then use the regular Lagrange basis: fenicsproject.org/olddocs/dolfin/1.6.0/python/demo/documented/biharmonic/python/documentation.html.

The general version of Poisson's problem is given at 3:04 with -u'' = f. Chossing f = -1 means that we have a uniform force acting in the entire domain, like gravity.