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.
21 questions with no upvoted or accepted answers
6
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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{...
3
votes
0
answers
74
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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 $$Y = \Phi(...
3
votes
0
answers
400
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Weak form of the Laplace-Beltrami operator on closed manifolds
Suppose we are dealing with diffusion over a boundaryless manifold $M$ (for simplicity let's say it's a surface). In that case, we have
\begin{align}
\int_M W \Delta U \mathrm{d} x & = -\int_M \...
2
votes
0
answers
52
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finding weak form of nonlinear differential equation for FEM simulation
The following is the well-known nonlinear differential equation for director's distribution at static equilibrium in liquid crystal displays(LCD). I want to obtain weak form of the given differential ...
2
votes
0
answers
386
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A general question about spectral methods vs finite element methods
According to this Wikipedia article:
Spectral methods can be computationally less expensive and easier to implement than finite element methods; they shine best when high accuracy is sought in simple ...
2
votes
0
answers
66
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Continuity of weak solution of elliptic PDE
I am investigating the following standard elliptic PDE with mixed Dirichlet-Neumann boundary condition:
$-\Delta u=f$ on $\Omega$; $u=0$ on $\Gamma_D$; $\left<n,\nabla u=g\right>$ on $\Gamma_N$
...
2
votes
0
answers
245
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Error estimates in the $L^2$ norm of conforming FEM about Poisson’s equation with mixed boundary conditions
Consider Poisson’s equation
$$- \Delta u = f{\rm\qquad{ in }}\;\Omega $$
with following mixed boundary cconditions
$$u = g{\rm\qquad{ on }}\;\Gamma \subset \partial \Omega $$
$$\frac{{\...
2
votes
0
answers
183
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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')...
2
votes
0
answers
79
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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}:...
2
votes
0
answers
927
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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,
$$
\left\{\!\!
\begin{aligned}
& -(a(x)u'(x))'+b(x)u(x)=f(x)\qquad x\in[0,1]\\
& u(0)=u(1)=0
\end{aligned}
\...
1
vote
0
answers
39
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Formulation of multipoint constraints using Lagrange multipliers for a time dependent problem (with the Finite Element Method)
Intro
Suppose we have the following static linear equations (e.g. of an elastostatic problem):
$$\mathbf{K}\boldsymbol{u}=\boldsymbol{f}$$
We want a multipoint constraint of the type
$$\boldsymbol{\...
1
vote
0
answers
63
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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 ...
1
vote
0
answers
47
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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 ...
1
vote
0
answers
80
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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 ...
1
vote
0
answers
62
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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 $\|\...
1
vote
0
answers
80
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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 $U=(u_1,u_2,...,u_N)\in\mathcal{H}$. ...
1
vote
0
answers
52
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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{...
0
votes
0
answers
24
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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 ...
0
votes
0
answers
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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 ...
0
votes
0
answers
88
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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 ...
0
votes
0
answers
139
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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{...