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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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Integration by parts & Finite Elements

I am reading on the Finite Element Method and I have the following question: For $f \in L^2(\Omega)$, $\sigma \in C^1(\Omega)$, find a finite element formulation of the problem \begin{equation} -\...
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How do you find the gradient of the edges (error estimation) in finite elements?

I am using the finite element method and need to find the errors associated with each of my elements. I am looking for help to find the error on the edges of the triangle, preferably by hand and not ...
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Quadratic finite element in 2d: Calculation of the right hand side

I'm trying to implement a solver for the 2d heat equation using FEM. I already having a working example for linear elements, there the approximation for the right hand side $f$ is calculated via a ...
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Understanding Finite element method

Suppose we have Poisson in 1D: $u'' = f(t)$ where $0<t<1$ and $u(0)=0$ and $u(1)=1$ We approximate the solution by $U(t) \approx \sum_{i=1}^n x_i \phi_i(t) $ where $\phi_i(t)$ are some basis ...
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Taylor in two variables - finite difference

I know that $U_{j\pm1}^n=u(x_{j\pm 1},t_n) \approx u\pm hu_x+\dfrac{h^2}{2!}u_{xx}\pm \dfrac{h^3}{3!}u_{xxx}+\mathcal{O}(h^4)$ and $U_{j}^{n+1}=u(x_{j},t_{n+1}) \approx u+ku_t+\dfrac{k^2}{2!}u_{...
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15 views

Stability of FEM using Crank-Nicolson

Let $V_h \subset \mathrm{H}'_0(\omega)$. $V_h$ is finite dimensional. The finite element with Crank-Nicolson for solving $$ \left\{ \begin{array}{ccc}{\frac{\partial P}{\partial t}-\nabla \cdot\left(\...
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Mixed Finite Element spaces

Consider the elliptic PDE $$-\frac{d^{2} u}{d x^{2}}=f(x)$$ where $x\in(0,1)$ and Dirichlet BCs are defined at both end points $$u(0)=0\text{ and }u(1)=0.$$ If we try to solve this using mixed finite ...
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What would cause Log-Log Plot to be Non-linear?

I'm attempting to plot the error of a simulation compared to an analytical solution and am curious if my problem is intrinsic to my model or the way I am plotting the results. The idea is that my ...
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Accuracy comparison Finite Difference, Finite Element & Boundary Element Method

I'm quite a newbie to numerical simulation (heat transfer) and I'm quite confused about a sentence that our teacher said. He said "Finite Difference Method (FDM) and Boundary Element Method (BEM) ...
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Is every linear finite element space over a bounded domain a subspace of the sobolev space H^1?

Since my knowledge of functional analysis, $L^p$-, Sobolev- and Hilbert spaces is not very good, I thought I could ask... Suppose we have a domain $\Omega \subset \mathbb{R}^2$ which is continuously ...
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Finite element solution for 1D Heat Equation Subject to both Neumann and Dirichlet BCs

I am trying to solve the heat equation $$ \frac{\partial u}{\partial t}-\kappa \frac{\partial^{2} u}{\partial x^{2}}=0 $$ with $$ u(0, t)=e^{-16 \pi^{2} \kappa t}, $$ $$ \frac{\partial u}{\partial x}(...
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Globalized Newton method for minimizing a specific functional: Convergence?

I'm currently working on a generalized p-Laplace equation: \begin{align} \label{DP} \begin{cases} \text{div} (\sigma |\nabla u|^{p-2} \nabla u) = f &\quad \text{in } \Omega\\ u = g &\quad \...
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the reference for block tridiagonal matrix of finite element discretization of 2D convection diffusion equation.

I need to know a block tri-diagonal matrix with Kronecker product structure arising from finite element discretization of 2-D convection-diffusion equation on square domain to test some codes. But I ...
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What is the difference between trial and test functions in the context of numerical integration?

I know that in, for example, Galerkin's method we try to approximate the solution via a sum $$ \sum_{j=1}^{n} u_{j} a\left(e_{j}, e_{i}\right) $$ with $a\langle \cdot ,\cdot \rangle$ a bilinear form ...
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Deriving the residual error for finite elements.

I am using the following set of notes for adaptive finite elements (https://www.ruhr-uni-bochum.de/num1/files/lectures/AdaptiveFEM.pdf) and am trying to go through the error calculations on page 29&...
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Relation between residue and closeness of solution for non-linear system $Ax=b$

Let $A(x)\in \mathbb{R}^{n\times n}$ be a matrix depending on $x$, $b\in \mathbb{R}^n$ such that, $$A(x)x=b,$$ i.e. we have a system of non linear equations. Let $u\in \mathbb{R}^n$ be its solution ...
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Multivariable Nonlinear equations

I am faced with the system depicted below We have a the following system: $$ [A]\left\{q\right\} + [B]\left\{q^2 \right\} + [C]\left\{ Q \right\} = \left\{ F\right\} $$ where $\left\{ q \right\...
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186 views

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{\...
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Finite element discretization example clarification

We introduce on the domain $\Omega = (0,1)$ a mesh $0=x_0<x_1<x_2<\dots<x_{n+1}=1$ and let $V_h$ be the space of piecewise linear hat functions $\varphi_i$ such that $$\varphi_i'= \...
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Most accurate mesh

Given the following problem, $$ (P) = \begin{cases} - \Delta u(x,y) = f(x,y) \qquad (x,y) \in \Omega \\ u(x,y) = 0 \qquad \forall (x,y) \in \partial \Omega \\ \end{cases} $$ (The ...
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48 views

Stiffness Matrix Formation for PDE with Neumann Boundary

Given the problem $$-\nabla u + u = f$$ $$ n\cdot\nabla u = g \quad\text{on} \quad \Gamma$$ I can show the discretization given through the Galerkin formulation is $Au=b$ where $$ A = \int_\Omega \...
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Solving a system of PDEs with only one Dirichlet boundary on one equation using FEM

I'm trying to solve a nonliner system of equations where there are 4 PDEs coupled through a nonlinear term j. The functions that I'm solving for are $\phi_s$, $\phi_e$, $c_s$, $c_e$, and $j$ using the ...
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66 views

Finite element heat equation on a single simplex?

I am currently trying to learn the finite element method. Ultimately, I want to solve the heat equation in arbitrary dimensions. For the purpose of this question, however, assume that I am interested ...
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Proof of norm of a solution of a differential equation is less than the norm of equation itself.

Consider the problem $$-eu''+xu'+u=f$$ $x$ is defined on the interval $I=[0,L]$. $u(0)=u'(L)=0$ where $e > 0$ is a constant. Prove that the solution satisfies $||eu''||≤ ||f||$ where norm is the $...
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Plotting 2D P2 Finite Elements solutions imported from FreeFem++ in Matlab

I really need your help, for a project I need to display in Matlab the solution to a PDE obtained in FreeFem++. I can plot any P1 solution but with higher order (which is required for my problem) it ...
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1answer
36 views

Direct sum factorization of polynomials

I have been recently reading the paper "Mixed finite elements for second order elliptic problems in three variables" by Brezzi et. al. I noticed the claim in the proof of Lemma $2.1$, which basically ...
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22 views

FDM variable coefficients

Usually authors of the books demonstrate the usage of FDM on the following equation $\frac{\partial f}{\partial t} - a\frac{\partial^2 f}{\partial x^2} = f(x,t)$ where $a$ is some constant. Is it ...
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Finite elements for biharmonic equation

I have a question regarding the conforming finite elements for biharmonic equation: if we want to discretize the weak formulation for this problem using a conforming finite element space $V^h$, is it ...
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60 views

Why the Galerkin Orthogonality Holds?

This is not homework. I'm going over my lecture notes to study for an exam. For an Abstract Elliptic Problem such as the problem with V a Hilbert Space $$\begin{cases} \text{Find } u \in V \text{ ...
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How do theoretical convergence speeds translate into real life when using FEM?

I'm currently solving the wave equation in a 1d and a 2d domain using the finite element method in space and the leapfrog or crank-nicolson method in time. Theoretically, I expect a convergence of $O(...
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68 views

How Young inequality was applied?

Let $|\cdot|$ a norm in $L^2$ and $\|\cdot\|$ a norm in $H_0^1$. Then $\begin{align} &|u_h^n|^2 + 2 \theta \Delta t h^{-1} |u_h^n|\|u_h^{n+\theta}\| |u_h^{n+ \theta} - u_h^n| +2 \theta \Delta t |...
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Finite Element Method for vector valued functions

I need help with the finite element method for the following problem I present in weak formulation. Certain details are left out since they are not important for the essence of this question. This is ...
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How to implement the Kutta condition for potential flow and FEM?

I have a potential flow FEM solver (basic Laplace equation) which works well and is validated for potential flow around cylinder and symmetrical airfoils (such as NACA0012). However, when the angle of ...
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FEM approach to obtain the projection of a function onto the mesh?

I need to explain the background of the problem I'm working on, and the questions themselves are a couple of paragraphs down. I am reading this book and implementing the algorithms along (so far in ...
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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 ...
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Show that the variational formulation has at most one solution

We have the problem: $$ -u''(x) + u(x) = f(x) ,\quad \quad x \in [0,L] $$ $$u(0) = 0 $$ $$u'(L) + u(L) = 4 $$ I then put it into variational form (hopefully correctly done) with introduction of $v \...
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Property of a quasi-uniform triangulation

I have some type of proof for the inverse inequality: $|\nabla v |_{H^1} \le C |v|_{H^1}$ This proof uses the following property for quasi-uniform triangulations: $ \frac{\int_{{K}^\wedge}{|\nabla v^\...
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Why does my newton-Raphson iteration fail?

Suppose that I have the following energy equation that is a function of $\varepsilon$, the strain, and $\eta$, the hardening parameter. $\phi=\frac{1}{2}E \varepsilon_e^2+\frac{1}{2}H \eta^2$, $\...
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FEM: Testing basis functions in a subspace V_h

In the finite element method, at a certain point we arrive at the following Galerkian problem where it is desired to find the solution $u_h \space \in V_h$ that solves the following equation: $$ a(...
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First derivative approximation using function values in equidistant points

Given a function $f: [x_0,x_4] → \Bbb R$ and equidistant points $x_0, x_1, x_2, x_3, x_4$ so that $h=x_{i+1} - x_i > 0$. Normally I would do, $f'(x) \approx \dfrac{f(x+h)-f(x)}{h}$, but here I ...
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Determining a variational formulation of $u^{(4)} = f$ with $3$-rd order BC.

Below is a problem from a recent exam and I have some questions about it. Given the boundary value problem:$$\dfrac{\partial^4 u}{\partial x^4} = f,\quad f\in L^2(0,1)$$ $$u(0) = u''(0) = u'(1) = u''...
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variational formulation of second order differential equation

I am given the following differential equation. Let $\Omega = (a,b)\subset\mathbb{R},\ f:\Omega \rightarrow\mathbb{R},\ \alpha,\beta \in \mathbb{R}$ and $$ -u'' + u = f \\ u(a)= \alpha, u(b) = \beta ...
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1D Finite element method: Function contineously differentiable?

I m currently reading the book "The finite element method in electromagnetics" written by Jian-Ming Jin. On page 50 and 51 which I attach as a screenshot below they show for an example functional, ...
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Finite element method for nonlinear differential equation

I encounter this problem $$\frac{df(u(x))}{dx} = g(x)$$ with $$u(0) = u(1) = 0$$ I first convert it to weak form $f(u(x))v(x)]^1_0 - \int^{1}_0 \frac{dv(x)}{dx}f(u(x))dx =- \int^{1}_0 \frac{dv(x)}{dx}...
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Calculate surface normal and area for a non-planar quadrilateral

Given the four coordinates of the vertices, what is the best possible approximation to calculate surface area and outward normal for a quad? I currently join the midpoints of the sides, thus ...
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Choice of the family of the Basis Functions

As I have learned for now, there are several families of Polynomial-type Basis functions (Lagrange, Serendipity, Hermite, ...). My question is beside the order of the elements which affects the order ...
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Why are differential equations satisfied/defined only on open domains?

I am going through FEM lectures by University of Michigan, and in one of the lectures, the professor writes an ODE: $$\frac{d^2u}{dx^2}+f(x)= 0, \ \ \ x \in (0,L)$$ Why are these differential ...
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Iteration step of the Crank–Nicolson scheme

For many dynamic problems a masse matrix $M$ has to be taken into account too. Consider a discretized systems of the form $$ \textbf{M}\frac{d}{dt}\vec{u}(t) = -\textbf{A} \cdot \vec{u}(t) + \vec{f}(...
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For a bounded linear functional $l(v) := (f,v)_\Omega$, do we have $\|l\| \le \|f\|$?

Suppose we have a Poisson equation $-\Delta u = f$ on $\Omega$ and we want to derive its weak formulation, so we multiply it by an arbitrary test function $\forall v \in H^1_0$ and then take integral ...
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finite element non linear boundary value problem

I have the following small finite element non linear boundary value problem: $$ -u''(x) = 1 + u^2(x) \quad \text{for} \quad 0<x<1 \quad \text{with} \quad u(0) = u(1) = 0 $$ with a grid with ...