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Accepted

• 56.2k

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
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
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

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

"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
Accepted

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
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
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

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

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
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

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

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

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
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

$O(h^2)$ 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

• 56.2k
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