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9 votes
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,...
Daniel Shapero's user avatar
7 votes
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 ...
Steven Roberts's user avatar
6 votes
Accepted

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

The Poisson equation with variable coefficient reads $$ - \nabla \cdot (\alpha \nabla u) = f. $$ After the usual integration by parts, the weak form is given as $$ \int_\Omega \alpha(x) \frac{\partial ...
Julian Roth's user avatar
6 votes

error estimate inequalities in finite elements

What the exact form of these error estimates is depends on a number of factors: What norm you measure the error in. When you state $e < h^p$, I assume you measure the error $e$ as the $H^1$ ...
Wolfgang Bangerth's user avatar
6 votes
Accepted

Continuous vs discontinuous space-time FEM

More concretely, it can be shown that discontinuous Galerkin (dG(r)) schemes lead to strongly A-stable time stepping schemes and continuous Galerkin (cG(r)) schemes are A-stable time stepping schemes, ...
Julian Roth's user avatar
6 votes

PhD in scientific computing to be a scientific programmer

I think you are asking the wrong question. You are asking "do I need a PhD for folks to hire me as a scientific programmer?" This is a kind of hypothetical. You're asking and getting ...
Richard's user avatar
  • 3,971
6 votes
Accepted

Keeping the surface flat for finite element analysis

You should never apply loads or boundary conditions on nodes in a FE model. Points have no physical meaning in the real world, at least when speaking of continua. Normally you would apply a load or ...
Konstantinos's user avatar
6 votes

Any FEM book recommendations that focus on stability and proofs on error bounds?

I would suggest the books below: Susanne C. Brenner , L. Ridgway Scott - The Mathematical Theory of Finite Element Methods Daniele Boffi , Franco Brezzi , Michel Fortin - Mixed Finite Element Methods ...
Abdullah Ali Sivas's user avatar
5 votes
Accepted

Formulation of $-\mathrm{div}(k\nabla u)=f$ in $\Omega$ for the Finite Element Method

If you take the functions given for $u$ and $k$ and apply the differential operator that you have in your differential equation you exactly obtain $f$, that is $$f = - \operatorname{div}(k \...
nicoguaro's user avatar
  • 8,524
5 votes

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 ...
Wolfgang Bangerth's user avatar
5 votes

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 ...
cos_theta's user avatar
  • 451
5 votes
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 ...
nicoguaro's user avatar
  • 8,524
5 votes
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 (...
lightxbulb's user avatar
  • 2,197
4 votes
Accepted

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 ...
Mikael Öhman's user avatar
4 votes
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{\...
Wolfgang Bangerth's user avatar
4 votes

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 ...
RandomElasticity's user avatar
4 votes
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 ...
whpowell96's user avatar
  • 2,636
4 votes

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 ...
Julian Roth's user avatar
4 votes

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 ...
Konstantinos's user avatar
4 votes

FEM textbooks recommendation

I like the book by Elman, Silvester, and Wathen, "Finite elements and fast iterative solvers" quite a lot.
Wolfgang Bangerth's user avatar
4 votes

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 ...
Wolfgang Bangerth's user avatar
4 votes

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 ...
Wolfgang Bangerth's user avatar
4 votes
Accepted

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 ...
Laurent90's user avatar
  • 1,943
3 votes

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 ...
Chenna K's user avatar
  • 944
3 votes
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 ...
Wolfgang Bangerth's user avatar
3 votes

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{...
HEMMI's user avatar
  • 186
3 votes

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 ...
Dohn Joe's user avatar
  • 559
3 votes

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 ...
manifoldcurious's user avatar
3 votes

in Finite Element, which approximation requires less number of unknowns: B-splines vs Shape functions vs Spectral Elements

Based on the comments below your post you may reach the conclusion that # DOFs and speed have no correlation whatsoever - this is not true. Keeping all other things fixed and increasing the number of ...
lightxbulb's user avatar
  • 2,197
3 votes

How to calculte the contributions of a lift operator in FEM?

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 $\...
Wolfgang Bangerth's user avatar

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