2
votes
2answers
105 views

dual of $H^1_0$: $H^{-1}$ or $H_0^1$?

I have a problem related to dual of Sobolev space $H^1_0$. By definition, the dual of $H^1_0$ is $H^{-1}$, which contains $L^2$ as a subspace. However, from Riesz representation theorem, dual of a ...
3
votes
0answers
55 views

estimations of solutions

Let $\Omega=\mathbb{R}^2_+=\{(x,y) \in \mathbb{R}^2; y > 0\}$, $f \in L^2(\Omega)$, $\lambda \in \mathbb{R}^*_+$, $A=(a_{ij})_{1\leq i,j \leq 2}$, $a_{ij} \in \mathbb{R}$ and there exist $\alpha ...
3
votes
1answer
94 views

Having trouble understanding the finite element method

I am trying to read some sources on how to implement the finite element method and I am having difficulty putting all the concepts together. I can read and understand the Galerkin approach just fine. ...
3
votes
0answers
75 views

In which space does my (weak) solution exist?

I am reading through some numerical analysis papers, and was wondering what characteristics of a problem define which function space the weak solution of a problem belongs to. For example, for the ...
0
votes
0answers
29 views

Prove $\|{v}\|_{W^{m,p}(T)} \le C|T|^{1/p - 1/q} \cdot h_{T}^{-m} \cdot \|{v}\|_{L^{q}(T)}$

My professor asked me to derive this inverse estimate: $\|{v}\|_{W^{m,p}(T)} \le C|T|^{1/p - 1/q} \cdot h_{T}^{l-m} \cdot \|{v}\|_{W^{l,q}(T)}$, for $l \le m$ So I divided the problem into 2 steps: ...
0
votes
1answer
35 views

Could the functions in larger space than $L^2$ be approximated by finite element basis functions?

Let $u \in V:=\{v \in L^{1+\alpha}(\Omega): \nabla v \in \mathbf{L}^2(\Omega)\}$, where $0<\alpha<1$. Clearly, $H^1(\Omega):=\{v \in L^2(\Omega): \nabla v \in \mathbf{L}^2(\Omega)\} \subset ...
2
votes
1answer
81 views

Density of smooth functions in fractional Sobolev space

I am reading a paper on the analysis of numerical methods, and am confused about a statement made. I am working in fractional Hilbert spaces, but I don't think that this has much bearing on the answer ...
1
vote
0answers
35 views

$H^1$ estimate by $L^2$-norm

Let $(\tau_h)$ be a shape regular triangulation. Prove that there exists a constant $c>0$ such that $$\|v\|_{H^1(\Omega)}\leq \frac{c}{\min_{T\in\tau_h}} \|v\|_{L^2(\Omega)}$$ for all $v\in V^h$ ...
-1
votes
2answers
117 views

finite elements-exercice

We consider in $\mathbb{R}^2$ the set of points $$\{M_1(-1,1),M_2(0,1), M_3(2,1),M_4(-1,0),M_5(1,0),M_6(2,0)\}$$ Let $\Omega$ a rectangular structure consisting of the heads $\{M_4(-1,0),M_6(2,0), ...
1
vote
0answers
30 views

Density of finite element functions in $W^{1,p}(\Omega)$

I would like to know if the following statement is true: For each $u \in W^{1,p}(\Omega)$ and $\varepsilon > 0$ there exists a piecewise affine function $u_{\varepsilon}$ and a triangulation of ...