All Questions
1,333 questions
57
votes
3
answers
39k
views
What are the conceptual differences between the finite element and finite volume method?
There is an obvious difference between finite difference and the finite volume method (moving from point definition of the equations to integral averages over cells). But I find FEM and FVM to be very ...
- 5,439
55
votes
5
answers
11k
views
What are criteria to choose between finite-differences and finite-elements
I am used to thinking of finite-differences as a special case of finite-elements, on a very constrained grid. So what are the conditions on how to choose between Finite Difference Method (FDM) and ...
- 3,720
33
votes
9
answers
6k
views
Modern resources for learning FEM
I need to get started using Finite Element Methods. I am about to start reading Numerical solutions of partial differential equations by the finite element method by Claes Johnson, but it's dated 1987....
- 1,739
30
votes
9
answers
2k
views
What is a good way to run parameter studies in C++
The problem
I'm currently working on a Finite Element Navier Stokes simulation and I would like to investigate the effects of a variety of parameters. Some parameters are specified in an input file ...
- 505
29
votes
3
answers
6k
views
What is the purpose of using integration by parts in deriving a weak form for FEM discretization?
When going from the strong form of a PDE to the FEM form it seems one should always do this by first stating the variational form. To do this you multiply the strong form by an element in some (...
- 511
26
votes
1
answer
11k
views
What are differences between 'a priori' and 'posteriori' error estimate in numerical analysis?
I have learnt about Finite Element Method (also a little on other numerical methods) but I don't know what are exactly definition of these two errors and differences between them?
- 801
25
votes
4
answers
4k
views
How to incorporate the boundary conditions with the Galerkin method?
I've been reading some resources on the web about Galerkin methods to solve PDEs, but I'm not clear about something. The following is my own account of what I have understood.
Consider the following ...
- 1,739
24
votes
9
answers
53k
views
Basic explanation of shape function
I just started studying FEM in a more structured basis compared to what I used to do during my undergraduate courses. I am doing this because, despite the fact that I can use the "FEM" in commercial (...
- 342
24
votes
1
answer
12k
views
What is the general idea of Nitsche's method in numerical analysis?
I know that the Nitsche's method is a very attractive methods since it allows to take into account Dirichlet type boundary conditions or contact with friction boundary conditions in a weak way without ...
- 801
19
votes
1
answer
6k
views
How to derive the Weak Formulation of a Partial Differential Equation for Finite Element Method?
I have taken a basic introduction to Finite Element Method, which did not emphasize a sophisticated understanding of a 'weak formulation'. I understand that with the galerkin method, we multiply both ...
- 12.2k
19
votes
1
answer
25k
views
How to formulate lumped mass matrix in FEM
When solving time dependent PDE's using the finite element method, for example say the heat equation, if we use explicit time stepping then we have to solve a linear system because of the mass matrix. ...
- 1,929
17
votes
2
answers
14k
views
In FEM, why is the stiffness matrix positive definite?
In FEM classes, it's usually taken for granted that the stiffness matrix is positive definite, but I just can't understand why. Could anyone give some explanation?
For instance, we can consider the ...
- 699
16
votes
3
answers
2k
views
Role of the numerical flux in DG-FEM
I am learning the theory behind DG-FEM methods using the Hesthaven/Warburton book and I am a bit confused about the role of the 'numerical flux.' I apologize if this is a basic question, but I have ...
- 672
15
votes
2
answers
8k
views
What is the purpose of the test function in Finite Element Analysis?
In the wave equation:
$$c^2 \nabla \cdot \nabla u(x,t) - \frac{\partial^2 u(x,t)}{\partial t^2} = f(x,t)$$
Why do we first multiply by a test function $v(x,t)$ before integrating?
- 301
15
votes
3
answers
6k
views
Finding which triangles points are in
Suppose I have a 2D mesh consisting of nonoverlapping triangles $\{T_k\}_{k=1}^N$, and a set of points $\{p_i\}_{i=1}^M \subset \cup_{k=1}^N T_K$. What is the best way to determine which triangle each ...
- 3,225
15
votes
2
answers
3k
views
FeniCS: Visualizing high order elements
I've just started messing around with FEniCS. I am solving Poisson with 3rd order elements and would like to visualize the results. However, when I use plot(u), the visualization is just a linear ...
- 542
15
votes
1
answer
3k
views
Visualizing discontinuous Galerkin/finite element data
I would like to visualize simulation results, obtained using the discontinuous Galerkin (DG) approach, within ParaView. Similarly to finite volume methods, the problem domain is divided into cube-...
14
votes
3
answers
20k
views
Alternatives to Comsol Multiphysics
This might be a question better suited for the Software Recommendations side of S.E., however I do believe that people who frequent this part of S.E. are more likely to be able to answer this question....
- 255
14
votes
6
answers
7k
views
What is a common file/data format for a mesh (for FEM)?
I'm developing an FEM simulation. For early testing, I will use simple self-written mesher and visualisation of the mesh graph. But I want to prepare my program to use data generated by an existing ...
- 1,463
14
votes
1
answer
463
views
What spatial discretizations work for incompressible flow with anisotropic boundary meshes?
High Reynolds number flows produce very thin boundary layers. If wall resolution is used in Large Eddy Simulation, the aspect ratio may be on the order of $10^6$. Many methods become unstable in this ...
- 25.8k
14
votes
2
answers
1k
views
Impose the compatibility conditions for mixed finite elements method in Stokes equation
$\newcommand{\v}[1]{\boldsymbol{#1}}$
Suppose we have following Stokes flow model equation:
$$
\tag{1}
\left\{
\begin{aligned}
-\mathrm{div}(\nu \nabla \v{u}) + \nabla p &= \v{f}
\\
\mathrm{div} \...
- 2,552
14
votes
0
answers
546
views
Sequential approach to solving coupled PDEs
I'm dealing with a coupled system of three transient, non-linear convection-diffusion equations. Let's just say to simplify the problem that they take the following form:
$$
-\nabla\cdot(D_{1}(u_{2},...
- 947
13
votes
3
answers
3k
views
Mathematically, why does mass matrix / load vector lumping work?
I know that people often replace consistent mass matrices with lumped diagonal matrices. In the past, I've also implemented a code where the load vector is assembled in a lumped fashion rather than ...
- 12.2k
13
votes
2
answers
2k
views
Which preconditioners (and solver) in PETSc for indefinite symmetric systems should I use?
My system is a symmetric FE problem with lagrange multipliers (e.g. incompressible Stokes' flow):
\begin{pmatrix}A & B^T \\ B & C\end{pmatrix}
where $C = 0$ is the typical case (I have even ...
- 996
13
votes
3
answers
1k
views
Finite elements on manifold
I'd like to solve some PDEs on manifolds, say for example an elliptic equation on a sphere.
Where do I start? I'd like to find something that use preexisting code/libraries in 2d , nothing so fancy (...
- 133
13
votes
5
answers
2k
views
Calculation of the sparsity structure for finite element matrices
Question: What methods are available to accurately and efficiently calculate the sparsity structure of a finite element matrix?
Info: I'm working on a Poisson Pressure Equation solver, using Galerkin'...
- 133
13
votes
1
answer
3k
views
What are the relative benefits of using Adams-Moulton over Adams-Bashforth algorithm?
I am solving a system of two coupled PDE's in two spatial dimensions and in time computationally. Since the function evaluations are expensive, I would like to use a multistep method (initialised ...
- 301
13
votes
1
answer
2k
views
What are possible methods to solve compressible Euler equations
I would like to write my own solver for compressible Euler equations, and most importantly I want it to work robustly in all situations. I would like it to be FE based (DG is ok). What are the ...
- 2,930
13
votes
2
answers
455
views
Oscillations in singularly perturbed reaction-diffusion problems with finite elements
When FEM-discretizing and solving a reaction-diffusion problem, e.g.,
$$
- \varepsilon \Delta u + u = 1 \text{ on } \Omega\\
u = 0 \text{ on } \partial\Omega
$$
with $0 < \varepsilon \ll 1$ (...
- 3,146
12
votes
3
answers
463
views
How to write integration tests for numeric simulation software?
Just to be more precise, I'll put a worthy example of my typical use case.
Let's say I'm developing a FEM software that produces several temporal solutions and inserts them in an HDF5 file, along with ...
- 223
12
votes
1
answer
8k
views
What is a "hanging node" in the finite element meshing?
When reading literatures about finite element method, the term "hanging nodes" can often be encountered. Could anyone tell me what indeed is a hanging node?
- 699
12
votes
1
answer
388
views
How to integrate polynomial expression over 3D 4-node element?
I want to integrate a polynomial expression over a 4-node element in 3D. Several books on FEA cover the case where integrating is performed over an arbitrary flat 4-noned element.
The usual procedure ...
- 501
12
votes
2
answers
631
views
Verification in Eigenvalue problems
Let us start with a problem of the form
$$(\mathcal{L} + k^2) u=0$$
with a set of given boundary conditions (Dirichlet, Neumann, Robin, Periodic, Bloch-Periodic). This corresponds with finding the ...
- 8,622
12
votes
1
answer
586
views
DG local equation, how to interpret mean-averaged test function
In the paper http://www.sciencedirect.com/science/article/pii/S0045782509003521, an HDG element-local equation is described on page 584 equation (4), with one of the equations taking the following ...
- 672
11
votes
5
answers
3k
views
Motivation behind Galerkin method
I have a question about Galerkin method. I don't understand why the Galerkin method weights the residual by the shape functions and sets it equal to zero. I want to know what is reason of this. Why we ...
- 111
11
votes
2
answers
7k
views
How to efficiently implement Dirichlet boundary conditions in global sparse finite element stiffnes matrices
I am wondering how Dirichlet boundary conditions in global sparse finite element matrices are actually implemented efficiently. For example lets say that our global finite element matrix was:
$$K =
\...
- 1,929
11
votes
3
answers
4k
views
Poisson equation: Impose full gradient as boundary condition via Lagrange multipliers
I have a physical problem governed by the Poisson equation in two dimensions
$$
-\nabla^2 u = f(x,y), \; in \; \Omega
$$
I have measurements of the two gradient components $\partial{u}/\partial{x}$ ...
- 213
11
votes
3
answers
9k
views
How to properly apply non-homogeneous Dirichlet boundary conditions with FEM?
In general, Dirichlet boundary conditions won't be satisfied exactly for FEM for non-homogeneous boundary conditions. The FEM codes I've seen set the degrees of freedom to interpolate the Dirichlet ...
- 476
11
votes
2
answers
7k
views
FEM: singularity of the stiffness matrix
I'm solving the differential equation
$$
\left( \sigma^{2}(x) u ''(x) \right)'' = f(x), \;\;\; 0 \leqslant x \leqslant 1
$$
with initial conditions $u(0) = u(1) = 0$, $u''(...
- 445
11
votes
1
answer
4k
views
Raviart-Thomas elements on reference square
I'd like to learn how the Raviart-Thomas (RT) element works. To that end I'd like to analytically describe how the basis functions look on the reference square. The goal here is not to implement it ...
- 625
11
votes
3
answers
1k
views
Best Methodologies for Managing a Mesh in Parallel Finite Element Computation?
I am currently developing a domain decomposition method for the solution of the scattering problem. Basically I am solving a system of Helmholtz BVPs iteratively. I discretize the equations using ...
- 111
11
votes
2
answers
4k
views
Discontinuous Galerkin / Poisson / Fenics
I am trying to solve the 2D Poisson equation using
the Discontinuous Galerkin method (DG) and the following
discretization (I have a png file but I am not allowed
to upload it, sorry):
Equation :
$$\...
- 115
11
votes
2
answers
1k
views
Space-time finite element discretization for time-dependent PDEs
In FEM literature, semi-variational methods are typically used in the solution of time-dependent PDEs. I have not seen a fully-variational approach i.e. where space and time are discretised by FEM, ...
stephn28
11
votes
1
answer
350
views
What are the possible numerical schemes for a diffusion equation with a nonlinear reaction term?
For some simple convex domain $\Omega$ in 2D, we have some $u(x)$ satisfying the following equation:
$$
-\mathrm{div}(A\nabla u)+cu^n = f
$$
with certain Dirichlet and/or Neumann boundary conditions....
- 2,552
10
votes
3
answers
18k
views
The real myth of GPU (specifically CUDA) really speed up FEM/CFD
Now I have been believing that FEM/CFD is supposed to be faster on a GPU unit - here I am using CUDA as solid example. However, I have not been able to find a convincing paper where the benchmark ...
- 475
10
votes
4
answers
2k
views
When we use Bernstein polynomials in application
When it is preferable to use Bernstein polynomials to approximate a continuous function instead of using the only following preliminary Numerical Analysis methods: "Lagrange Polynomials", "Simple ...
10
votes
3
answers
8k
views
What is the difference between implicit FEM and explicit FEM?
What is the difference between explicit FEM and implicit FEM exactly? According to the post here, it seems that the only difference is whether implicit or explicit time integration is used.
As I ...
- 335
10
votes
3
answers
3k
views
Are there any "light-weight" FEM packages around?
Basically, FEM seems to be a problem that is pretty much "solved". There are numerous powerful frameworks existing, like Trilinos, PETSc, FEniCS, Libmesh or MOOSE.
One thing they have in common: They ...
- 1,463
10
votes
2
answers
328
views
Finite elements $W^{1,\infty}$ error estimates
Are there finite element method setups that provide error estimates in the $W^{1,\infty}$ norm (i.e., bounds on $\|u'_h - u'\|_\infty$)? Which families of elements can be used for implementing them?
(...
- 12.6k
10
votes
1
answer
2k
views
$L^2$-convergence of finite element method when right hand side is only in $H^{-1}$ (Poisson eqn)
I know that the piecewise linear finite element approximation $u_h$ of
$$
\Delta u(x)=f(x)\quad\text{in }U\\
u(x)=0\quad\text{on }\partial U
$$
satisfies
$$
\|u-u_h\|_{H^1_0(U)}\leq Ch\|f\|_{L^2(U)}
$...
- 809