# 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.

339 questions
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### 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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### 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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### 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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### 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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### Use trace theorem to define $H^2_0$ space and the requirement of the boundary?
For homogeneous biharmonic problems, the solution space is in general defined as $$H^2_0(\Omega):=\{u\in H^2(\Omega): u=\frac{\partial u}{\partial \mathbf{n}}=0\text{ on }\partial \Omega\}.$$ With ...