14
votes
Accepted
$L^2$-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
$$\|u\|_{L^2} = \sup_{\phi\in L^2\...
- 12.4k
6
votes
Accepted
FEM current toy problem
The formulation of this problem is tricky. Here is what you have in your original post:
Find $h \in L^2(\partial D)$ such that for any $w \in H^{\frac 1 2}(\partial D)$,
$$
\int_{\partial D} hw \,...
- 56.2k
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 ...
- 2,104
5
votes
Accepted
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 ...
- 2,104
5
votes
Accepted
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 ...
- 128
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 ...
- 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 ...
- 56.2k
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 $\Delta u$ ...
- 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 ...
- 1,157
4
votes
Accepted
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 ...
- 56.2k
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 ...
- 56.2k
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 ...
- 10.4k
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 ...
- 2,269
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 Fréchet derivative of the differential operator.
...
- 451
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 ...
- 6,229
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 ...
- 8,582
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 ...
- 459
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 ...
- 121
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
$...
- 173
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, ...
- 906
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 ...
- 3,197
2
votes
Accepted
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'...
- 56.2k
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 ...
- 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):
$$ \tag{$*$}\label{eq:poisson}
\frac{d^2 \phi}{dx^2}(x) = f(x), \quad 0<x<1, \...
- 831
2
votes
Simulating the heat equation with insulating material
Why even bother to simulate the left half? Why not just simulate the right half with the left BC being the constant value? Seems simpler and more accurate to what you want to actually model, because ...
- 2,099
2
votes
Index reduction of a DAE from a PDE system
The usual approach to solving these kinds of system is not to actually think of them as an ODE with an algebraic constraint, but to use specialized algorithms that makes use of the structure. The ...
- 56.2k
2
votes
Accepted
Simple to program method for elliptic PDE with curved boundary?
For complex domains like what you describe, I think that the finite element method is basically the only way to go.
Spectral discretizations are nice when the spatial domain is simple, but your ...
- 10.4k
2
votes
Accepted
Galerkin projection in AMG
Sorry, I have been busy lately. I don't have enough time to write the answer I want to, however, here is the short one: In GMG you define the prolongation and restriction operators based on the finite ...
- 2,851
1
vote
Accepted
FEM solution for Poisson is not exact at nodes
The finite difference scheme also gives exact solution at the nodes to the problem $-u''=1$ because
$$
\frac{u_{j-1} - 2 u_j + u_{j+1}}{h^2}
$$
is exact for a quadratic function.
For the more general ...
- 3,038
1
vote
Morley element implementation reference
I have implemented Morley elements for the biharmonic equation according to Kirby's paper:
arxiv.org/abs/1706.09017
I was specifically interested in the 2D problem:
$$\Delta^2 u(x) = 0, x \in \...
- 2,352
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