Questions tagged [finite-element-method]

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.

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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?
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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…
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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…
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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
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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…
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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
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Understanding Finite element method

Suppose we have Poisson in 1D: $u'' = f(t)$ where $0
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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…
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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
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Courant (1943) and History of Finite Element Method

I am interested in the history of Finite Element Methods and Methods of Weighted Residuals (MWR), especially reduced quadrature and collocation methods. I have a paper coming out called “Orthogonal Collocation Revisited” which has a brief section on…
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Lagrange multiplier for the Stokes equations

I'm trying to understand the following: Let $\Omega \subset \mathbb{R}^2,\ V$ = space of (vectorvalued) continuous piecewise linear functions with zero boundary, $W = $ space of continuous piecewise linear (scalar) function. We're looking for…
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How can i understand this contradiction of $H^{-s}(\Omega)$

This makes me puzzled.Here $H^{k}(\Omega)$ is the sobolev space $W_p^k(\Omega)$ with $p=2$. We all know that $H^{k}(\Omega)$ is Hilbert space,This means that $(H^k(\Omega))^{*}=H^{-k}(\Omega)$ should be $H^{k}(\Omega)$ itself from Risez…
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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
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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…
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Finite Element Method for the 1d wave equation

I'm solving the 1D wave equation \begin{equation} \frac{\partial^2 \eta}{\partial t ^2} - \frac{\partial^2 \eta}{\partial x ^2} = 0 \end{equation} with boundary conditions \begin{equation} \frac{\partial \eta}{\partial x} = 0 \qquad \qquad \text{on}…
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