# Tag Info

Accepted

### Does the weighted residual method not use energy minimization in any form?

The answer to your question depends on the problem. For example, consider the diffusion equation for a field $q$, with diffusivity $k$ and sources $f$. The variational form of this problem states that,...
• 10.3k
Accepted

### Does this second-order implicit Runge-Kutta method have a name?

As Wolfgang stated in the comments, this is not a traditional RK due to the inconsistent time evaluations within a stage. At first it would seem it can't even be cast as an additive RK since terms ...
• 1,114
Accepted

• 8,524

### Query about FE approximation of a Poisson equation with non-constant coefficients

@julianroth already gave the right answer, namely that the equation is typically of the form where there is only one derivative on the coefficient. The situation would be more complicated if the ...
• 55.7k

### Why can the weak forms of the Stokes and continuity equations be combined into a single equation?

To get from weak form I to II, just add the two equations of weak form I. To see the other direction, note that weak form II has to hold for all test functions $\phi$ and $\psi$. That is, you can test ...
• 451
Accepted

### Quintic Hermite shape functions

I mentioned how to do it for cubic polynomials in a previous answer. And has an expanded version in my blog. You can do the derivation in global coordinates and obtain the following global basis ...
• 8,524
Accepted

### "Cleanest, most professional" approach to implement a finite differences scheme in dimension greater than two

Finite differences are implemented by fitting a (multivariate) interpolating polynomial through a set of points and taking the derivatives of said interpolating polynomial. Contrary to popular belief (...
• 2,197
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### Dirichlet condition in finite element method

From a different perspective, regardless of where a system $A\cdot x = 0$ comes from (does not have to be FEM), if you change your mind and would like to prescribe part of $x$, then you'd necessarily ...
• 1,008
Accepted

### Can a mixed boundary conditions in 1D Linear FE lead to non positive definite stiffness?

Let's combine your $\alpha,\beta,\gamma$ into fewer symbols by diving your boundary conditions by $\alpha$. For the general-dimensional case,  -\Delta u = f, \qquad \text{in $\Omega$}, \\ \frac{\...
• 55.7k

### Why does the choice of basis functions change the approximate solution in FEM even when the same space is spanned?

Expanding the discussion in the comments, The change of basis does not matter under the same space $V_h$, you end up with the same solutions whichever basis you use. This can be seen easily for ...
Accepted

### Solving of KU=F leads to numpy.linalg.LinAlgError: Singular matrix

It looks as if you are making entire rows and columns of the stiffness matrix zero. This makes the matrix singular. If the solution's value is fixed at some entry, then remove that row/column from the ...
• 2,636

### How to generate mesh for space-time FEM method in FEniCS?

First of all, for FEniCS specific questions please refer to the official FEniCS forum. In my opinion, your question can be answered irrespective of your FEM library of choice. There are two ...
• 786

### How to compute overall inertia properties from FE mass matrix?

Although this is not a direct answer to your question, what one should know here is how inertia moment calculations can be performed by FE in general. For this purpose and in order to avoid trivial ...
• 381

### FEM textbooks recommendation

I like the book by Elman, Silvester, and Wathen, "Finite elements and fast iterative solvers" quite a lot.
• 55.7k

### Volume change of a deformable cylinder with a uniform spinning angular velocity

You can take an analogy of trying to simulate a circular orbit of a single particle in a gravitational potential. If you use the explicit Euler method, for example, your orbit will become larger and ...
• 55.7k

### Constructing metric terms for high order elements

Let's assume you have a way of computing the 27 tri-quadratic shape functions associated with the 27 points (either by tabulating them as polynomials, or by computing them as interpolating Lagrange ...
• 55.7k
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### How to refine $h$ and $\Delta t$ for convergence tests on evolution PDE

Here are a few solutions that you could explore to determine the orders in space and time. 1) Separate study of time error You can use a given spatial mesh, and perform multiple simulations with finer ...
• 1,943

### A confusion about the bubble function in lumped mass FEM

For the P2b element, that is, the quadratic triangle with a cubic bubble, all the basis functions contain the term $\xi _{1} \xi _{2} \xi _{3}$, where $\xi_3=1-\xi_1-\xi_2$. Thus, each basis function ...
• 944
Accepted

### what preconditioner for incompressible hyperelasticity in 3d (similar to stokes equation?)?

There is no difference between the linear and nonlinear Stokes problems as far as preconditioning is concerned. That is because at the end of the day, you always have to linearize the nonlinear ...
• 55.7k

### Can a mixed boundary conditions in 1D Linear FE lead to non positive definite stiffness?

You are right, positive definiteness is NOT always true. I checked it again for a simple case. As a simple example, consider a 1-dimensional Laplace equation for the region $0\leq x \leq 1$: \begin{...
• 186

### PhD in scientific computing to be a scientific programmer

If your goal is to become a programmer at the company X that creates the scientific software Y, then sure: do a PhD in a field that uses software Y. Several of my colleagues who used and extended the ...
• 559

### H(curl) conforming Nédélec-Elements to satisfy div(B)=0

The answer you are looking for is found in the notion of weak divergence. Recalling some basic facts about distributions, we say that a function $u$ has a weak divergence if for any smooth function ...
The lift operator takes a function defined on the boundary of a cell and produces a function that lives in the interior of the cell. You will evaluate the bilinear form $B_h$ for basis functions \$\...