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Questions tagged [l2-norm]

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1
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0answers
70 views

computing dual matrix trace norm and tensor gradient in python

I'm trying to write the following function in python: $$ f_\mu(\mathcal X) = f_0(\mathcal X) + \sum_{i = 1}^n \max_{||\mathcal Y_{i(i)}|| \leq1} \alpha_i\langle \mathcal X_{(i)},\mathcal Y_{i(i)} \...
5
votes
2answers
126 views

Residual norm of PDE discretization: correspondence in the continuous problem?

Solving a linear PDE like $$ \Delta u = f \quad\text{on } \Omega,\\ n\cdot \nabla u = 0 \quad\text{on } \Gamma, $$ with Finite Elements usually goes like this: Create the discretization $Au=b$ via $$ ...
1
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0answers
119 views

Estimate $L_2$ norm of a elliptic problem with unknown exact solution on finite element method

I have the elliptic problem $$-\Delta u = 1,\,\,\Omega\subset\mathbb{R}^2$$ with $u=0$ on $\partial\Omega,$ with $\Omega=[-1,1]^2\backslash([0,1]\times[-1,0])$ and I want to estimate the $L_2$ error ...
4
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1answer
187 views

$L^2$-error in FEM: how to compute integral over reference element?

I have the following problem. The domain is $(0,1)$ and we consider a uniform triangulation on $\hat{\Omega}$ with elements $K_i = [i/N,(i+1)/N]$ and $X_h^1$ the linear finite element space. I wrote ...
0
votes
1answer
224 views

Physical interpretation of L2 norm of heat equation solution

For the heat equation \begin{equation} u_t(t,x) = \nu u_{xx}(t,x) \end{equation} for $x \in [0,1]$ with boundary conditions $u(t,0) = u(t,1) = 0$ and initial value $u(0,x) = u_0(x)$ it is easy to ...