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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Change of variables via Jacobian with Voigt notation for differential operator
I'm reading Non-linear Finite Element Analysis of Solids and Structures (De Borst, R., Crisfield, M. A., Remmers, J. J., & Verhoosel, C. V).
As part of the weak form of the governing equations of ...
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How to prove this inverse inequality about boundary flux when using FEM?
I'm currently studying the article by Pehlivanov et al.:
Pehlivanov, A. I.; Lazarov, R. D.; Carey, G. F.; Chow, S. S., Superconvergence analysis of approximate boundary-flux calculations, Numer. Math. ...
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How can the Finite Element Method be used to solve a Multi-equation System?
Every explanation I've seen of FEM show it solving an equation of the form $\mathcal{L}\phi=f$, with $\mathcal{L}$ being a differential operator, and every solution (whether it be Ritz or Galerkin) ...
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What does this text mean in the context of the approximating solution for method of weighted residuals?
I am going through some introductory material on spectral/hp element methods and the book has a section on method of weighted residuals. I am paraphrasing the contents of the book below.
If we ...
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Variational formulation of the vector Laplace equation in cylindrical coordinates
I want to solve the vector Laplace equation $\nabla^2 \mathbf{v}=\mathbf{f}$ in arbitrary coordinate systems using finite-elements.
The usual way to derive the variational form necessary for the ...
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intersection of kernels for bilinear forms
I have
$$
b_{\Omega_i} (u,v) = a_{\Omega_i} (u,v) - k^2(u,v)_{\Omega_i}
$$
and
$$
(\Xi(u), \Xi(v)_ {1, k, \Omega_i} = a_{\Omega_i} (u,v) + k^2(u,v)_{\Omega_i}.
$$
Where
$$
a_{\Omega_i}(u,v)= \int_{\...
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How to transform an integro-differential equation into weak form for FEM
I have the following 2D boundary-value problem which I would like to solve numerically using FEM software:
\begin{equation}
a(x,y)\nabla^2 u(x,y) + \int\int K(x,y,x',y')u(x',y') dx' dy' = f(x,y),
\end{...
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Is there always a sequential nature of time-stepping? [closed]
Pretend someone is solving a partial differential equation using something like the finite element method. Lets say they are calculating the propagation of a seismic wave through inhomogeneous terrain ...
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Error in the energy norm for inhomogeneous Dirichlet boundary conditions
Short description
When conducting FEM analysis with inhomogeneous Dirichlet boundary conditions, I compute the error in the energy norm with an expression that should only work for problems with ...
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How to use local approximation spaces to build a global space via the Partition of Unity Method
I'm working to understand how the partition of unity method is used to build a global PUM space from local approximation spaces. Can someone please explain the mechanics of gluing the local spaces ...
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Weak formulation of the biharmonic equation
The biharmonic equation is
$\Delta^2 u = 0$.
Tutorials show that the stiffness term in the weak formulation is:
$\int \langle\Delta \varphi_i, \Delta \varphi_j\rangle$
However, FEM implementation uses
...
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Making sense of a PDE for linear elasticity: gradient of a vector field
Example 2 here presents a linear elasticity problem with the weak form
$$
-{\rm div}({\sigma}({\bf u})) = 0
$$
where
$$
{\sigma}({\bf u}) = \lambda\, {\rm div}({\bf u})\,I + \mu\,(\nabla{\bf u} + \...
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Weak formulation for $-u''+u=1$
I am trying to solve the following problem.
Now, based on other examples I have seen, it seems the weak formulation should be something in terms of an inner product. In particular, we see
$$(-u+1,v)=(...
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linear elasticity equation and spectral equivalence
Consider
$$\nabla \cdot \sigma = f \quad \text{in} \quad \Omega \\
\sigma = 2\mu\varepsilon(u) + \lambda ( \nabla \cdot u ) I \quad \text{in} \quad \Omega \\
u = 0 \quad \text{over} \quad \Gamma_D \\...
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Uniform equivalence of grid functions
I am studying the finite element method and have found a problem:
For a regular and quasi-uniform family of simplicial partitions, prove uniform equivalence
$H^1$ norms of grid functions to the norm:
$...
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1
answer
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Static solution to an implicitly dynamic problem - heat equation
Heat equation
This is the heat equation:
$
\frac{\partial u}{\partial t}
= \frac{\partial^2 u}{\partial x^2}
+ \frac{\partial^2 u}{\partial y^2}
+ \frac{\partial^2 u}{\partial z^2}
$
...
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1
answer
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L2−conforming (discontinuous) vs integration points
Finite element discretization spaces
Full de Rham complex
The picture below taken from here, displays from left to right:
H1−conforming (continuous)
H(curl)−conforming (continuous tangential ...
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Computing $L^2$ norm and quadrature for time-varying solution to PDE?
Setup. On domain $\Omega \times (0,T]$, the parabolic problem
\begin{align}
u_t + \Delta u + u = f
\end{align}
with some appropriate initial and boundary conditions has solution $u(x,t)$, which I ...
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Small 2D-FEM Problem: Heat Transfer from Soil to Water Pipe
I want to create a simple FEM-Model for a geothermal system from scratch.
At first I'd like to formulate the PDE and then the weak form for a simplified 2D problem. I'm not that good at maths and new ...
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1
answer
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Weak form FEM discretisation of Non-Linear System
I am trying to derive the FEM solution for the equation $u''(x) + \left( u'(x) \right)^2=0$ with $u(1)=0, u'(0)=1$ over the interval $[0, 1]$.
Constrain the trial function, $v$, to also be zero at $x=...
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Implementing natural boundary conditions for the heat conductivity of a rod
I'm facing the following problem regarding the heat conductivity of a rod, $t: (0,1) \rightarrow \mathbb{R}$
$-(c(x)t'(x))'=h(x),$
$t(0)=0,$
$c(1)t'(1)=d$
With $x \in (0,1)$, $d \in R$, $c(x)$ and $h(...
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In finite element method for second order elliptic problem with neumann boundary value, is the solution weakly satisfies the boundary conditions?
For example, let's consider the problem
\begin{equation}
-\Delta u+u = 0
\end{equation}
For $g\in H^{\frac{1}{2}}((D)$, where $D$ is the domain, assume that $u$ is the solution of
\begin{equation}
(u,...
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Implementing Boundary Conditions at Infinity in Numerical Method
Suppose I want to numerically solve a differential equation, the analytical solution of which has boundary conditions at infinity, for example
$$
\tag{1} \begin{cases}
u_t - \nabla^2 u = 0 \quad (\...
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Is there an H^1 conforming interpolation with boundary condition with estimate?
Recently I have been researching Finite element analysis and I need an interpolation that satisfies (interpolation space $X_h\subset P^k$)
boundary interpolation $(u,v_h)_{\partial\Omega}=(\Pi u,v_h)...
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lagrange interpolation estimate in finite element analysis
It suddenly occurred to me that when we apply the Bramble-Hilbert lemma to estimate the error, we have something like
\begin{equation}
||v-\Pi v||_0\leq Ch^{k}|v|_{k}
\end{equation}
where the ...
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Two equivalent statements yielding different values when evaluated?
Assume $v_i$ and $v_j$ are tent functions.
We have the expression:
$$ \int^b_a v_iv_j'' dx $$
Integrating by parts gives:
$$ \int^b_a v_iv_j'' dx = -\int^b_a v'_j v_i' dx + [v_iv_j']^b_a $$
Now let'...
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Galerkin Method - Why does integration-by-parts eliminate need to enforce Neumann boundaries?
Disclaimer: I have not taken any class in functional analysis and the only class I have ever taken in differential equations is at the barest-bones introductory level.
From what I know of the Galerkin ...
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What is the correct convergence rate for my PDE?
Let $G = (0,1)$ and $J = (0,1)$ and consider the PDE
\begin{cases}
\frac{\partial u}{\partial t} - \frac{\partial^2 u}{\partial x^2} &= f(t,x) \quad \text{on } G \times J, \\
u(x,0) &= u_0(x) \...
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Second order elliptic PDE with inhomogeneous Dirichlet boundary condition
For second order elliptic PDE, we usually consider the weak solution, which is the solution for variational form. When facing inhomogeneous Dirichlet boundary conditions, we usually modify it into ...
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Weak formulation for the forward Kolmogorov equation
I want to solve the Forward Kolmogorov equation by Finite element method:
$Lf = (\nabla f)^T b + \frac{1}{2}tr((\nabla^2f)D)$
where $D = \begin{pmatrix}
D_{11} & D_{12}\\
D_{21} & D_{22}
\end{...
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Weak form of 1D linear elliptical PDE, integration by parts
Please refer to the attached image.
I am struggling to understand how the integration by parts is performed for the weak form.
Can anyone please assist with an explanation?
The equation is not written ...
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answer
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FEM for non linear PDEs
I am looking for an easy but rigorous reference for FEM methods for non linear PDEs like the p-laplace equattion or non linear heat equation ect.
Can one recommend me a good exposition of this topic ...
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Simple block factorization?
In the two substructures case for the finite element tearing and interconnecting method (FETI) from "Domain Decomposition: Parallel Multilevel Methods for Elliptic Partial Differential Equations&...
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Why is easier to get inverse of mass matrix?
On my lecture notes, I came across the statement: 'Inverting the mass matrix is significantly easier than solving a linear system involving the stiffness matrix.'
Can someone explain why this is the ...
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1
answer
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Determine Functions of a Reference Triangle for Finite Element
I am given a problem to define the functions $\phi_1(x, y)$, $\phi_2(x, y)$ and $\phi_3(x, y)$ for a single triangular element as a reference to do element assembly. However I'm not sure why the ...
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Customizing the bump function
I have the standard bump function below.
$$ \Psi(x) = e^{-\frac{1}{1 - \mathrm{min}(1, x^2)}} $$
How can I customize it to be like below:
Translate and scale
I can translate and scale by:
$$ \Psi(x) ...
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Basis functions that are one at a point and zero at others
1D basis functions
In 1D space, I'm looking for a set of basis functions like below. They would be equal to one at certain points and zero at others.
2D schematics
In 2D space, the basis function ...
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How to add other constraints on boundary to enforce the pde not to be identically 0 in the domain
I need to solve a stationary density distribution of a Fokker Planck equation
$-\nabla \cdot(\mu \nabla V) + \frac{1}{2}\sigma^2 \Delta\mu = 0 $ where $\mu$ is the stationary density of the SDE:
$dX = ...
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1
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How to solve a boundary value problem with Neumann Boundary conditions using the Finite Element Method
I have been given the question:
Consider the boundary value problem:
$$-u'' + u = f(x) \forall x \in (0,1), u(0) = 0, u'(1) = 0.$$
Using continuous piecewise linear basis functions on a uniform mesh
...
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Seeking name of "trick" involving operators like $A + \tau B$, where $B$ is Lipschitz.
Theorem.
On Hilbert space $V$, suppose $T: V \to V$ is nonlinear and that $T = A + \tau B$ where $A$ is linear and strongly monotone, $B$ is nonlinear and Lipschitz, and $\tau > 0$ can be made ...
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Mixed Laplace equation and missing one step in the integration by parts
I am looking at the Deal.ii finite element package docs, and was reading the section about vector valued problems. These are simply systems of PDEs that are solved together.
I am trying to derive the ...
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Finite element method: what are bilinear and linear forms?
Sample problem
Consider this ODE:
$$ u : \Omega \to \Re $$
$$\nabla u-u=0 \qquad \mathrm{in} \qquad \Omega$$
$$\Omega\subset\Re$$
$$ u = 1 \qquad \mathrm{on} \qquad \Gamma$$
$\Omega\subset\Re$ is the ...
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How to solve this coupled PDE eigenvalue problem numericallly?
I'm solving a system of two coupled partial integro-differential equations for two functions $ {\phi _0^\alpha (R,r')} $ and $ {\phi _0^\beta (R,r)} $:
$$ \frac{{{\partial ^2}\phi _0^\alpha }}{{\...
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Why Applying A Continuous Linear Elliptic Differential Operator Gives A Functional and Clarifying Dual Space Definition
I'm reading a paper on Multiscale FEM Methods and I just need a bit of help better visualizing how to interpret the dual space in the PDE setup.
We are given a continuous, linear, elliptic ...
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0
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variational formulation of a mixed boundary condition problem and prove the existence of solution
This is a mixed boundary value problem:
\begin{align*}
-\nabla \cdot(\nabla u) &= f, & x &\in \Omega \\
u &= g_D, & x &\in \Gamma_D \\
\frac{\partial u}{\partial \...
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0
answers
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Difference between serendipity and Lagrange elements in the Finite Element Method
As an engineer I often use the finte element method to solve solid mechanics problems, both by commercial software and by my own (matlab) code. When using quadrilateral and hexahedral elements there ...
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0
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Why minimizing the functional form give the solution of BVP
Consider the linear BVP, $$-\frac{d}{dx}\left(p(x) \frac{dy}{dx}\right)+q(x)y=f(x),\quad 0<x<1, y(0)=y(1)=0$$
We assume that $p(x)\geq\delta>0$ for some constant $\delta$, and $q(x)\geq0$ on $...
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votes
0
answers
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purpose of interpolants in Galerkin methods
I am learning about the finite element method in an abstract Banach and Hilbert space setting, especially for an application to differential forms, and I am a bit lost on the big picture about the ...
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votes
0
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inhomogeneous Helmholtz equation does not obey superposition using FEM
I am trying to solve the in-homogeneous Helmholtz equation with damping and forcing using finite element method (FEM). The equation is;
$c^2\nabla^2p - c^2\omega i d \nabla^2p + \omega^2 p = f$
where ...
1
vote
1
answer
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Essential Boundary Condition and Natural Boundary Conditions in Weak Form Galerkin
Some days ago my teacher gave me a question about using weak form Galerkin to solve an ODE. I'm not so good at solving Differential Equations so I'm here asking for some help(By the way my English is ...
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