53

Finite Element: volumetric integrals, internal polynomial order Classical finite element methods assume continuous or weakly continuous approximation spaces and ask for volumetric integrals of the weak form to be satisfied. The order of accuracy is increased by raising the approximation order within elements. The methods are not exactly conservative, thus ...


32

I've always found the approach to describing finite element methods that focuses on the discrete linear system and works backward unnecessarily confusing. It is much clearer to go the other way, even if that involves a bit of mathematical notation in the beginning (which I'll try to keep to a minimum). Assume that you are trying to solve an equation $A u = ...


28

Nitsche's method is related to discontinuous Galerkin methods (indeed, as Wolfgang points out, it is a precursor to these methods), and can be derived in a similar fashion. Let's consider the simplest problem, Poisson's equation: $$ \left\{\begin{aligned} -\Delta u &= f \qquad\text{on }\Omega,\\ u &= g \qquad\text{on }\partial\Omega. \end{aligned}\...


25

Error estimates usually have the form $$ \|u - u_h\| \leq C(h),$$ where $u$ is the exact solution you are interested in, $u_h$ is a computed approximate solution, $h$ is an approximation parameter you can control, and $C(h)$ is some function of $h$ (among other things). In finite element methods, $u$ is the solution of a partial differential equation and $...


22

When I studied the finite element method in graduate school, this notion of multiplying by a weight function was also very alien to me. Eventually, I did find a nice (albeit non-rigorous) analogy that helped me understand it. This analogy is based on 3D vector geometry and an understanding of projections and dot-product. 3D Geometry Imagine a 2D plane ...


18

Short answer: No, you don't have to do integration for certain FEMs. But in your case, you have to do that. Long answer: Let's say $u_h$ is the finite element solution. If you choose piecewise linear polynomial as your basis, then taking $\Delta$ on it will give you an order 1 distribution (think taking derivative on a Heaviside step function), and the ...


17

Nothing stops you from doing that technically, but when you integrate by parts you get more flexibility with the solution space in that they need not have $H^2$ regularity (required for the non I.B.P formulation). The linear elements you suggest generally have enforced continuity between elements, and so could not be in $H^2$. The I.B.P formulation ...


17

The following picture illustrates a mesh with a hanging node and a mesh containing no hanging node: Usually with a finite element mesh the vertices are shared with their other neighbouring elements, but the circled node does not belong to the bottom triangle. We call this node a hanging node. This commonly occurs during the process of adaptive mesh ...


17

The property follows from the property of the corresponding (weak form of the) partial differential equation; this is one of the advantages of finite element methods compared to, e.g., finite difference methods. To see that, first recall that the finite element method starts from the weak form of the Poisson equation (I'm assuming Dirichlet boundary ...


16

The usual randomized edge hopping method should work. Basically, start with any triangle of the mesh, then determine which of the edges the target point lies on the opposite side of. That is, determine which of the edges, when extended out to a line, separate the point from the interior of the triangle. When there are two possibilities, choose one at random, ...


16

In the finite element method, the matrix entries and right hand side entries are defined as integrals. We can, in general, not compute these exactly and apply quadrature. But there are many quadrature formulas one could choose, and one often chooses them in a way so that (i) the error introduced by quadrature is of the same order as that due to ...


15

Excellent answers already on this page, but there is still a (small) missing point. The OP asked: Now, let's say that I have a PDE with higher order derivatives, does that mean that there are many possible variational forms, depending on how I use Green's formula? And they all lead to (different) FEM approximations? Integrating by parts (in the ...


14

Chapter 8 of Brenner and Scott's Mathematical Theory of Finite Element Methods is devoted to this subject. In particular, theorem 8.1.11 and the corollary give you that $ \|u - u_h\|_{W^1_\infty} \le C h^{k - 1}\|u\|_{W^k_\infty}$ for linear elliptic problems with sufficiently smooth coefficients, provided that the finite element space satisfies some other ...


14

Here's the deal with GPUs. On a GPU, every single core is slow. Really slow. However, you have thousands of cores. If you can effectively use the thousands of cores at a time, then your algorithm will run better on the GPU. If you cannot, then it will run much slower on the GPU. Linear algebra is one domain where parallelism is really well established. ...


13

The CFL condition states that the "mathematical domain of dependence" must be (asymptotically) contained in the numerical domain of dependence. For hyperbolic problems, this provides a bound $\Delta t < C \Delta x$ that is useful at all resolutions. For a parabolic problem, it merely requires that $\Delta t \in o(\Delta x)$ in the limit $\Delta x \to 0$. ...


13

Warning Solving saddle point problems involves a lot more choices than definite problems, and there are a lot more things that can go wrong. Use monitors for all levels to debug convergence, to sure that null spaces are handled correctly when auxiliary operators are singular (usually just a constant null space), and to ensure that preconditioners are ...


13

I do not think that there is a definite answer to this, because it might change from one topic to other (and also depends on the type of elements you are using). There are some recent papers talking about that, as well [2]. So, it is not a closed discussion. Furthermore, you can have different inertial components (at least in mechanics), when you have ...


13

You're coming at it backwards. The justification is better seen by starting from the variational setting and working towards the strong form. Once you've done this, the concept of multiplying by a test function and integrating can then be applied to problems where you don't start with a minimization problem. So consider the problem where we want to ...


12

You can interpolate the solution onto a finer mesh and then plot it: from dolfin import * coarse_mesh = UnitSquareMesh(2, 2) fine_mesh = refine(refine(refine(coarse_mesh))) P2_coarse = FunctionSpace(coarse_mesh, "CG", 2) P1_fine = FunctionSpace(fine_mesh, "CG", 1) f = interpolate(Expression("sin(pi*x[0])*sin(pi*x[1])"), P2_coarse) g = interpolate(f, ...


12

Adams-Moulton method is significantly more stable. The analogy used when I was taught the difference is the same as extrapolation and interpolation. Interpolation is relatively safe numerically. Extrapolation can blow up if you happen to have an asymptote or some other odd feature. For instance, solving the ode $y'(t) = -y(t)$ with $y(0) = 1$ using ...


12

A perfectly matched layer (PML) is generally introduced as a continuous concept without reference to the discretization method. A damping term is multiplied by the advection operator: $$ \frac{\partial}{\partial x} \rightarrow \frac{\partial}{\partial x}\left(\frac{1}{1 + i\sigma/\omega}\right) $$ If $\sigma$ is positive, the imaginary component in the above ...


12

Yes, this is the standard Aubin-Nitsche (or duality) trick. The idea is to use the fact that $L^2$ is its own dual space to write the $L^2$-norm as an operator norm $$\|u\|_{L^2} = \sup_{\phi\in L^2\setminus\{0\}} \frac{(u,\phi)}{\|\phi\|_{L^2}}.$$ We thus have to estimate $(u-u_h,\phi)$ for arbitrary $\phi\in L^2$. To do that, we "lift" $u-u_h$ to $H^1_0$ ...


11

If it's shock-capturing that you're interested in, I would suggest you use the finite volume method instead of the finite element method. When applied naively, FEM is actually notoriously bad at resolving shocks -- usually there are spurious oscillations or unwanted diffusion. Provided your original PDE is a conservation law, the FVM method will preserve ...


11

The Finite Element Method (FEM) is the parent method which has inspired many, many other methods and methods which are actually FEM but pretend not to be. In the finite element method, "shape functions" are used to provide an approximation space so that the solution can be represented by a vector. In the classical FEM, these shape functions are polynomials. ...


11

Not sure if you find the COMSOL Model Wizard somewhere else, maybe other commercial Multi-physics software but not in the open-source community. I had the same question a couple of years ago and I listed all Finite-element, Multi-physics framework. As you may know there are many of them. The one that I found really useful and close, at least in the way that ...


11

In general, you cannot just transfer the same polynomial basis from tetrahedral to quadrilateral elements.1 In particular, the whole point of quadrilateral elements is to work with tensor products of one-dimensional polynomials, which is not possible for tetrahedral elements. There are in fact quadrilateral Raviart-Thomas elements, but their definition is ...


11

As I mention before, I prefer to think about the weak form as a weighted residual. We want to find an approximate solution $\hat{u}$. Let us define the residual as $$R = c^2 \nabla \cdot \nabla \hat{u} - \frac{\partial^2 \hat{u}}{\partial t^2} - f(x,t)$$ for the case of the exact solution the residual is the zero function over the domain. We want to find ...


11

In deal.II (http://www.dealii.org -- disclaimer: I'm one of the principal authors of that library), we do eliminate whole rows and columns, and it is not too expensive overall. The trick is to use the fact that the sparsity pattern is typically symmetric, so you know which rows you need to look into when eliminating a whole column. The better approach, in ...


11

Suppose that the solution $u$ of the PDE lives in some function space $X$. We'll write the PDE as a bilinear form $A(u, v) = f(v)$ for all $v$ in $X$, where $f$ is some element of the dual space $X^*$. To approximate the solution $u$ of this PDE, we can instead look for some field $u_N$ that lives in a finite-dimensional subspace $V_N$ of $X$. Typically, ...


11

Your example is a pretty good indication that the two derivatives (with respect to $x$ and with respect to $u$) do not commute :) (In fact, they're very different beasts -- one is a Fr├ęchet derivative, the other a weak derivative.) Rather, you should consider $a(u) = (\partial_x u)^2$ as the composition $a = f\circ g$, of two functions $$ f: v(x)\mapsto v(...


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