A method of obtaining (numerically) approximate solutions to (usually) differential equations. It consists of a method of discretization splitting the domain into disjoint subdomains over each of which the problem has a simpler (approximate) solution, and a method of reassembling those pieces to obtain a solution over the whole domain. It is closely tied to the calculus of variations.
Questions tagged [finite-element-method]
665 questions
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votes
7 answers
What is the difference between Finite Difference Methods, Finite Element Methods and Finite Volume Methods for solving PDEs?
Can you help me explain the basic difference between FDM, FEM and FVM?
What is the best method and why?
Advantage and disadvantage of them?
Anh-Thi DINH
- 573
23
votes
1 answer
Variational formulation of Robin boundary value problem for Poisson equation in finite element methods
So I am confused about some details of obtaining a variational formulation specifically for Poisson's equation. I am in a Scientific Computing class and we just started discussing FEM for Poisson's equation in particular. However, we have a bunch of…
Fractal20
- 1,569
12
votes
1 answer
Finite Element Method Weak Formulation
I have a question about the weak formulation of a PDE in finite element analysis.
Suppose we have the following two-dimensional PDE:
$$ \Delta \cdot u(x,y) = q(x,y)
$$
where $q$ is given, $u$ is unknown, and $\Delta$ is the Laplacian operator…
kathleen
- 223
12
votes
5 answers
How to approximate numerically the gradient of the function on a triangular mesh
Given an arbitrary (lets say 2D) triangular mesh, with known $(x_i,y_i)$ locations of points, and numerical values of a function $f$ on them (either in the nodes, or in the centroids of the triangles, doesn't matter) like this random example, how…
Ander Biguri
- 552
7
votes
2 answers
How does Calculus of Variational work in Finite Element Method
I'm learning Finite Element Method. And it is said in a lot of books that Calculus of Variational is the basis of Finite Element Method. But as far as I know, Calculus of Variational is to find a function $f$ which will make the functional…
maple
- 3,023
7
votes
1 answer
Symmetric formulation for the heat equation
Consider the heat equation: $$\partial_t u-div(A\nabla u)=f$$
with $u(0)=0, u=0$ on the boundary of the domain of definition, call it $U$. Consider a test function $v=v(x,t)$, and perform the following operations:
integrate the strong formulation…
Lilla
- 2,177
7
votes
1 answer
Convert a general second order linear PDE into a weak form for the finite element method.
Problem
I want to convert the general second order linear PDE problem
\begin{align}
\begin{cases}
a(x,y)\frac{\partial^2 u}{\partial x^2}+b(x,y) \frac{\partial^2 u}{\partial y^2} +c(x,y)\frac{\partial^2 u}{\partial x \partial…
AzJ
- 924
7
votes
3 answers
Quadrilateral Interpolation
The simplest finite element shape in two dimensions is a triangle.
In a finite element context, any geometrical shape is endowed with an interpolation,
which is linear for triangles (most of the time), as has been explained in
this answer…
Han de Bruijn
- 17,475
6
votes
1 answer
Understanding Finite element method
Suppose we have Poisson in 1D: $u'' = f(t)$ where $0
ILoveMath
- 10,999
6
votes
1 answer
Finite Element on surfaces: evaluate solution
While working with a finite element for a PDEs solver on Riemannian Surfaces embedded in $\mathbb R^3$, I got stuck when needing to evaluate the solution $u$ at a given point $(x_0,y_0,z_0)$
The surfaces is approximated through a triangular mesh.
In…
mariob6
- 402
6
votes
3 answers
Finite element method books
I know this question has been asked before; I just want to enquire if anybody has any suggestions to learn how to compute finite element problems, including plenty of examples.
The topics I would like to focus in are as follows:
Introduction to…
MKF
- 632
6
votes
2 answers
What is the purpose of the weight function $w(x)$ in a Finite Element Method?
I have just started looking into finite element methods.
Suppose we have an equation for the strong
$$L(u) = s$$
Then the integral form of the equation is given by
$$\int_0^1 L(u)w(x) dx = \int_0^1 sw(x)dx$$
and "the choice of the weight function…
sonicboom
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votes
0 answers
Functions of fractional-order Sobolev spaces
In fracture mechanics one might end up dealing with functions such as
$$w(r,\theta) = \sqrt{r} \sin \frac \theta 2,$$
which is defined, for example, on a cracked unit circle, $\Omega = B(0,1)\setminus\{(x,0)\,|\,-1\leq x < 0\}$, and where…
knl
- 1,355
5
votes
2 answers
Is broken Sobolev space a Sobolev space?
The definition of a broken Sobolev space is as follows.
Given infinite-dimensional (but mesh-dependent)
spaces on an open bounded domain $\Omega \in R^3$ with Lipschitz boundary. The mesh, denoted by $\Omega_h$, is a disjoint partitioning of…
Jeff
- 51
5
votes
2 answers
Principle of virtual work and weak form
In finite element method, one wants to derive a so-called weak form of the differential equation to solve. This latter is obtained by multiplying both sides of the equation by a "test function" then apply the Green theorem.This test function is…
altprc
- 51