# 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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### 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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### Quadrature error estimates for $n$-rectangular finite elements in the context of elliptic second order problems

In Ciarlet's book "Finite Element Methods for Elliptic Problems" from 1978, in Chapter 4.1 "The Effect of Numerical Integration", the following Theorem is stated and proved: #######...
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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
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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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### 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 ...
1 vote
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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} ...
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{... 2 votes 1 answer 209 views ### 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 R^n, if I know that the boundary's of both are very close, how can I bound the determinant of the Jacobian between them. I ... 3 votes 0 answers 62 views ### 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? 1 vote 0 answers 50 views ### 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{... 1 vote 1 answer 392 views ### 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 ... 3 votes 0 answers 361 views ### 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 \... 0 votes 1 answer 112 views ### 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 ... 2 votes 0 answers 75 views ### 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}:... 1 vote 1 answer 794 views ### 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 ... 2 votes 0 answers 915 views ### 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} \...
$\DeclareMathOperator\diam{diam}$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{...