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14 votes
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$L^2$-convergence of finite element method when right hand side is only in $H^{-1}$ (Poisson eqn)

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\...
Christian Clason's user avatar
6 votes
Accepted

FEM current toy problem

The formulation of this problem is tricky. Here is what you have in your original post: Find $h \in L^2(\partial D)$ such that for any $w \in H^{\frac 1 2}(\partial D)$, $$ \int_{\partial D} hw \,...
Wolfgang Bangerth's user avatar
6 votes
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How do I avoid divide-by-zero when solving the Poisson equation with Fourier transforms?

You are solving Poisson's equation: $$\nabla^2 \Phi = 4\pi G \rho.$$ Notice that if $\Phi$ is a solution, then so is $\Phi+C$ for any constant $C$. Furthermore, $C$ will have no effect on your ...
David Ketcheson's user avatar
6 votes

How to compute numerically the $H^{1/2}$ norm of a function

I think what you are referring to is $\|h^{-1/2}g\|_{0,\partial D}$. The point is that it can be though of as the 'discrete' $H^{1/2}$ norm. It comes down to the so called 'inverse inequalities' where ...
knl's user avatar
  • 2,104
5 votes

biharmonic equation

I decided to expand my earlier comment into an answer. I'd suggest using the Morley element that uses $P_2$ basis in each element and the degrees of freedom are values at vertices normal derivatives ...
knl's user avatar
  • 2,104
5 votes
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How to correctly define the flux in a finite volume method for Poisson's equation with a piecewise constant material

Taking the average will certainly work, but is recommended to take the harmonic mean to account for the changing conductivity: $k = \Big( \frac{1-f}{k_i} + \frac{f}{k_{i+1}} \Big)^{-1}$ If you ...
Rover's user avatar
  • 128
5 votes
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Morley element implementation reference

I can write my experiences here because I do not have any book references at hand. Consider a triangular element with the corner points $\boldsymbol{x}_i \in \mathbb{R}^2$, $i=1,2,3$. The degrees of ...
knl's user avatar
  • 2,104
5 votes

"Optimal" domain partitioning in domain decomposition algorithms

We use domain decomposition because we want to exploit the power of more than one processor. As a consequence, the right question to pose is: "How do we need to partition the domain so that we ...
Wolfgang Bangerth's user avatar
5 votes
Accepted

Gradient jump in weak formulation

Let $\Omega = \Omega_1 \cup \Omega_2$, where $\{ \Omega_i \}$ represents two elements that share an interface $\Gamma = \Omega_1 \cap \Omega_2$. Assume that $u \in H^2(\Omega)$. Multiplying $\Delta u$ ...
Will P.'s user avatar
  • 831
4 votes
Accepted

$O(h^2)$ convergence for Elliptic PDE

You appear (and correct me if I am wrong, i do not wish to presume) to be confused on the notion of what is an order of convergence. The order of convergence is the order at which your approximate ...
BlaB's user avatar
  • 1,157
4 votes
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C or fortran library to solve linear 2D/3D elliptic PDE

Most of the widely used finite element libraries are written in C++. If all you really care for -- and if all you will ever care for -- is solving an elliptic PDE on a rectangle, then it's probably ...
Wolfgang Bangerth's user avatar
4 votes

Simple to program method for elliptic PDE with curved boundary?

One can certainly solve problems with curved boundaries using the finite difference method, but it is awkward. It is simpler to use the finite element method. There, you need to subdivide your domain ...
Wolfgang Bangerth's user avatar
4 votes

Poisson equation with discontinuous variable coefficient

First, some comments on discretization schemes for elliptic problems in general. There are some improvements you can make on the difference scheme that you've derived. The idealized problem is to ...
Daniel Shapero's user avatar
3 votes
Accepted

Finite-volume method applied to a particular advection equation

Your proposed discretisation appears to be consistent, but wouldn't normally be interpreted as a finite volume discretisation. Indeed, it looks a lot more like a finite difference method with a ...
origimbo's user avatar
  • 2,249
3 votes

Correctly setting boundary condition for periodic linear elasticity problem

Here is a description of a small FE model that might approximate the case of an infinite number of holes in an infinite plate. Create a model of a single repeating element with $1/4$ of a hole ...
Bill Greene's user avatar
  • 6,074
3 votes

Three steps of pde numerical solution and nonlinear equation

Linearization and discretization can be switched. The linearization then happens in function space, where a Newton method can be employed based on the Fr├ęchet derivative of the differential operator. ...
cos_theta's user avatar
  • 451
3 votes
Accepted

Finite element error for second order ODE at nodes equal to zero

For this simple elliptic PDE, the finite-element approximation is indeed exact at the nodes (i.e., coincides with the true solution); this is usually called superconvergence (at the nodes). Since it'...
3 votes
Accepted

Does a mixed method solve this elliptic pde exactly if the source function is piecewise polynomial?

The conjecture is false. Consider just the simplest case, $$ -\Delta u = 1 $$ in $\Omega=[0,1]^2$, and assume that $\partial u/\partial n=0$ all around (and fix the mean value of $u$ to make the ...
Wolfgang Bangerth's user avatar
3 votes

Semi-infinite domain transformation

You could use the following transformation \begin{align} &u = \tanh(x)\, ,\\ &v = \tanh(y)\, . \end{align} Another option is to use $2/\pi \arctan(x)$, but I have had better results with the ...
nicoguaro's user avatar
  • 8,515
3 votes
Accepted

Semi-infinite domain transformation

I know two papers that investigate infinite mapping layers and apply them to examples: [1] Schoder, Stefan, et al. "Revisiting infinite mapping layer for open domain problems." Journal of ...
Pepe's user avatar
  • 459
2 votes

biharmonic equation

The variational or weak formulation of the biharmonic problem reads as : find $u \in V_D$ such that $$ a(u,v) = \ell(v), \quad \forall v \in V_0,$$ where the bilinear and linear forms are given by $...
uli.xu's user avatar
  • 173
2 votes

Stokes Equation in "two-fold saddle point" form?

There is also some work on this in the literature on least-squares finite element methods, in which everything is reformulated as a first-order system; see for example this paper by Bochev and ...
Daniel Shapero's user avatar
2 votes
Accepted

Stokes Equation in "two-fold saddle point" form?

You can for instance have a look at the different HDG papers by Cockburn et. al. They use a similar form to formulate their hybridizable approaches. An overview is given in this review paper: http://...
Christian Waluga's user avatar
2 votes
Accepted

In mixed elliptic formulation, what are the weakest requirements to ensure the flux is in $H^1$?

I don't think you get this from your setup. If $\sigma$ is only in $L^2$, then the equation $\nabla \cdot q = \sigma$ would suggest that you only get $q \in H_\text{div}$, regardless of what boundary ...
Wolfgang Bangerth's user avatar
2 votes

$O(h^2)$ convergence for Elliptic PDE

I think if you are interested to find the order of convergence then you should vary the mesh spacing ($h$). Now as you vary the mesh spacing you can obtain the error of the numerical solution. Then if ...
Brato's user avatar
  • 121
2 votes
Accepted

How to impose boundary condition with mixed derivatives?

That's basically two questions in one. The first is how to incorporate boundary conditions, the other how to treat the symmetry. I'll consider only the symmetry here, because the boundary value stuff ...
davidhigh's user avatar
  • 3,147
2 votes

Raviart Thomas Mixed Finite Element with Mixed boundary conditions reference request

In this case, the essential boundary condition is Neumann and natural is Dirichlet. Since Neumann is essential and is nonhomogeneous, you can use the concept of lift, referred by a couple of authors, ...
likask's user avatar
  • 906
2 votes
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Stabilization parameter for an elliptic equation

The problem you want to solve, with a small $\kappa$ is "singularly perturbed", i.e., it generally has boundary and internal layers. The way you can think of it is that if $\kappa$ were zero, then you'...
Wolfgang Bangerth's user avatar
2 votes
Accepted

Imposing zero mean condition in FEM

There are two approaches to solve this. The first approach is given by Lagrange multipliers and is well explained here. You have to solve an extended linear system where you impose the solution to be ...
G. Gare's user avatar
  • 136
2 votes

Elliptic PDE: Proving that a 2nd order accurate discretization of the 2nd derivative of the unknown is 2nd order accurate for the unknown itself

Consider as a simple case the 1D Poisson problem (it should be simple to extend to the 2D case you describe): $$ \tag{$*$}\label{eq:poisson} \frac{d^2 \phi}{dx^2}(x) = f(x), \quad 0<x<1, \...
Will P.'s user avatar
  • 831

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