Questions tagged [finite-element-method]
The finite element method is a popular method for approximating numerically on a computer the solution of partial differential equations. It is based on a variational (weak) formulation of the PDE, followed by discretization on a finite-dimensional ansatz space which reduces the problem to a sparse linear algebra one.
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Imposing perfect slip boundary conditions in Stokes equations with Nitsche's method
I am working with a 2d mesh, given by a square with a circular hole placed in the middle. Let me call $\partial \Omega$ the boundary at the circle. I have a PDE for a vector field $\vec{v}$, and I ...
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What is the standard notation for bilinear, biquadratic, etc... spaces?
A typical notation for the polynomials of degree $k$ is $P_k$. The space $P_k$ is considered well-suited for interpolation on simplices, although that is hard to put into practice in full generality.
...
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Lumped mass matrices and bubble functions for tetrahedral elements
For whatever reason, I stubbornly decided to use tetrahedral elements and find myself needing to use P3 elements with bubble functions ("P3b3d" in FREEFEM-style denomination).
The 2d case is ...
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Coonvergence rate for the clamped plate problem when approximating with polygonal domains
I'm trying to understand Ridgeway Scott's "A survey of displacement methods for the plate bending problem" [1]. In chapter four he is talking about polygonal approximation and states that ...
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Formulation of multipoint constraints using Lagrange multipliers for a time dependent problem (with the Finite Element Method)
Intro
Suppose we have the following static linear equations (e.g. of an elastostatic problem):
$$\mathbf{K}\boldsymbol{u}=\boldsymbol{f}$$
We want a multipoint constraint of the type
$$\boldsymbol{\...
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Nitsche's method for p-Laplace equation
My question is about how you impose Dirichlet boundary conditions for the p-Laplace equation.
The minimization form of this problem is to find the function $u$ in $W_1^p(\Omega)$ that minimizes the ...
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How to solve with FEM a semilinear elliptic equation?
I searched in many books regarding FEM how to solve semilinear elliptic equation, but I did not find too many things. They mostly treat linear and simple problems. For example in P.Ciarlet-The finite ...
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Is Stokes equation a saddle point problem or a minimum problem?
Consider $\Omega \subset \mathbb{R}^2 $ (or $\mathbb{R}^3$). The well known stationary Stokes equations in the incompressible case are
\begin{equation}
\begin{cases}
- \Delta u + \nabla p = f \text{ ...
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Confusion with implementation of PDE constraint Bayesiain inverse problem
Consider a PDE,
$$\partial_t u -a \nabla u - ru (1-u) = 0$$
at a given snapshot in time. The inverse problem is to find the diffusion coefficient $a \in L^{\infty}$ from a noisy measurement $$Y = \Phi(...
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finding weak form of nonlinear differential equation for FEM simulation
The following is the well-known nonlinear differential equation for director's distribution at static equilibrium in liquid crystal displays(LCD). I want to obtain weak form of the given differential ...
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Pressure integrated by parts in finite element method
Most FEM texts or tutorials apply FEMs on diffusion equations where the 2nd spatial derivative is integrated by parts during weak formulation. For convection diffusion equations, there is a first ...
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Holomorphic "quasi-interpolation" of a function sequence
I am interested in some sort of analytic interpolation. A toy version of my problem is as follows. Let $V \subset \mathbb{C}$ be a complex neighborhood of $[0,1]$. Assume there is some bounded ...
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A general question about spectral methods vs finite element methods
According to this Wikipedia article:
Spectral methods can be computationally less expensive and easier to implement than finite element methods; they shine best when high accuracy is sought in simple ...
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Typo in error a-priori estimate in a discontinuous Galerkin paper?
I'm looking at this famous paper which is available in the link below:
Franco Brezzi, LD Marini, Endre Süli, Discontinuous Galerkin methods for first-order hyperbolic problems, Mathematical Models ...
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Difference between variation and differential
Been trying to understand how the numerical formulation for structural elements used in FEM are derived. Came across this piece from "Fundamentals of FEM for Heat and Fluid Flow" by Roland ...
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Is it desirable to establish a CAD-like geometry medium for four dimensional space-time topologies and FEM? [closed]
Finite element and numerical methods over 4d space-time is a topic of interest for elastodynamics right now. Do you think it is desirable to establish a geometry/CAD system for space-time involving ...
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Continuity of weak solution of elliptic PDE
I am investigating the following standard elliptic PDE with mixed Dirichlet-Neumann boundary condition:
$-\Delta u=f$ on $\Omega$; $u=0$ on $\Gamma_D$; $\left<n,\nabla u=g\right>$ on $\Gamma_N$
...
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Boundary integration of weak form in FEM using DG elements
If we use CG elements (continuous Galerkin), the boundary integration in FEM can be easily converted to sum over quadrature points using node basis functions of the edges. However, in DG elements (...
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Generate a two-variable polynomial from its "roots [closed]
I know, from mathematics basis, that a polynomial with one variable can be factor in function of its roots, so I can generate any one-variable polynomial from its zeros.
But I want know if is ...
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Error estimates in the $L^2$ norm of conforming FEM about Poisson’s equation with mixed boundary conditions
Consider Poisson’s equation
$$- \Delta u = f{\rm\qquad{ in }}\;\Omega $$
with following mixed boundary cconditions
$$u = g{\rm\qquad{ on }}\;\Gamma \subset \partial \Omega $$
$$\frac{{\...
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Unclear inequality of L2 norms (Poisson equation for modeling flow)
I encountered a problem working through a paper about modeling flow with the use of the Poisson equation (source given below). There appears an inequality of L2 norms I don't understand so far. Your ...
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Galerkin finite element method for solving third order time dependent partial differential equation in the weak form
How to solve the third order time dependent partial differential equation
$$u_t + 6u_x + u_xxx = 0$$
in weak form using galerkin finite element method?
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stability of parabolic problems where nonhomogeneous term in $L^2(0,T; H^{-1}(\Omega))$
There is a problem in this estimate. In Part 2, We can't derive the second inequality from the first inequality in Part 2. The main reason is the last term on the RHS in the first inequality is in $\|\...
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Solve 4th order ODE with variable coefficients
I am trying to solve a 4th order boundary value problem with variable coefficients, namely the problem of a rotating cantilever beam:
$u'''' - \frac{((1-x^2)u')'}{2\eta} - \frac{\alpha}{\eta}((1-x)u')...
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Estimate a projection from a product space of $H^1(\mathbb{R}^3)$ to a finite dimensional space
Let $\mathcal{H}=(H^1(\mathbb{R}^3))^N$ be the product space with the associated norm
$$
\Vert U\Vert_1=\left(\sum^N_{i=1}\Vert u_i\Vert_1^2\right)^{1/2}
$$
where $U=(u_1,u_2,...,u_N)\in\mathcal{H}$. ...
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How to solve Poissons equation using FEM with integral BC?
I have a 3-dimensional domain D, with 3 types of BC which I am trying to solve the Poisson equation on. $$-\nabla \cdot (\sigma(x,y,z)\nabla u)=0$$
Insulator (Neumann BC)
Electrode set at some ...
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Finite Element Method and Dirac eigenvalue problem
Consider Dirac equation in 2D with Hamiltonian given by (arb. units)
\begin{equation}
H=-i \begin{pmatrix}
0&\partial_x-i\partial_y\\
\partial_x+i\partial_y & 0\\
\end{pmatrix}.
\end{equation}
...
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Time discretization of the variational formulation of the Navier-Stokes equation
Let
$T>0$
$I:=(0,T]$
$d\in\mathbb N$
$\Lambda\subseteq\mathbb R^d$ be nonempty and open, $$\mathcal V:=\left\{\phi\in C_c^\infty(\Lambda,\mathbb R^d):\nabla\cdot\phi=0\right\}$$ and $$V:=\overline{...
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Bounding the determinant of the Jacobian between a set and its polyhedral approximation
My question is, essentially, suppose I have two simply connected subset of $\mathbb{R}^n$, if I know that the boundaries of both are very close, how can I bound the determinant of the Jacobian between ...
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Compatible Finite Elements [closed]
I have a basic question as a matter of definition. I am wondering what is meant by compatible finite elements? Does it has to do with the spaces over which the trial functions are defined?
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Approximation error estimate
I would like to find a good reference for the following or a similar, probably well-known, approximation error result:
Let $\Omega\subset \mathbb{R}^d$ be bounded, $p\in [1,\infty]$, $l, m\in \mathbb{...
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Generalized Lax-Milgram for Weak Formulation of 1D Linear Schrodinger
I am interested in the variational formulation of the 1D Schrodinger equation:
$i u_t- \beta u_{xx} = 0 $ and $u(x,0)=u_0(x)$ which upon integration by parts yields:
$i(u_t,v) + \beta (u_x,v_x) = 0$ ...
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Weak form of the Laplace-Beltrami operator on closed manifolds
Suppose we are dealing with diffusion over a boundaryless manifold $M$ (for simplicity let's say it's a surface). In that case, we have
\begin{align}
\int_M W \Delta U \mathrm{d} x & = -\int_M \...
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The classical two phase Stefan problems
What is the most commonly used treatment method of the moving interface in the classical two phase Stefan problems with the finite element method. Here I mean the water-ice two phase problem under ...
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Sobolev spaces, Finite Element Error Analysis
Let $\Omega\subset \mathbb{R}^{2}$ be a bounded, convex, polygonal domain and $H=\{(u_{1},u_{2})\in H^{\epsilon-\frac{1}{2}}(\Omega)\times H^{\epsilon-\frac{1}{2}}(\Omega),0<\epsilon<\frac{1}{2}:...
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Cea's lemma and norms
I would like your help understanding this article.
Page 239 (3.2 A priori error estimates), I am quickly getting lost because of the type of norm that is always changing.
Things I do not ...
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Comparison of cubic Hermite finite element and cubic B-spline finite element (regarding condition number of stiffness matrix)
Background:
Consider the one-dimensional second-order elliptic PDE,
$$
\left\{\!\!
\begin{aligned}
& -(a(x)u'(x))'+b(x)u(x)=f(x)\qquad x\in[0,1]\\
& u(0)=u(1)=0
\end{aligned}
\...
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Finite element method inverse estimate
$\DeclareMathOperator\diam{diam}$Looking for a proof in the literature of the following lemma:
Let $K\subset\mathbb{R}^d$ be a bounded domain. Let $P_X$ be a finite dimensional subspace of $\mathcal{...
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