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latex formatting improvement, question readability improvement, deleted unnecessary sentences at the end.
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edit approved May 16, 2017 at 18:05
Anton Menshov ♦
edit approved May 16, 2017 at 18:05
Anton Menshov ♦
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What's the definition of L^$L^{\infty}$-norm for nonconforming finite element?

We know that \begin{align*} \|u\|_{0,\infty,\Omega}={ess \sup}_{x\in \Omega}|u|. \end{align*}\begin{align*} \|u\|_{0,\infty,\Omega}={\text{ess} \sup}_{x\in \Omega}|u|. \end{align*} Moreover, the nonconforming Crouzeix-Raviart finite element space is \begin{align*} V_h=\{ v|_K\in P_1(K), v ~is~continuous~at~midpoints~of~edges~and~v=0~at~the~midpoints~on~boundary~edges\}. \end{align*}\begin{align*} V_h=\{ v|_K\in P_1(K),\ v ~\text{is continuous at the midpoints of edges and } v=0 \text{ at the midpoints on boundary edges}\}. \end{align*} For a function $w\in V_h$, we know that $w\notin H_0^1(\Omega)$. 

My question is what is the definition of $\|w\|_{0,\infty,\Omega}$ and $\|\nabla w\|_{0,\infty,\Omega}$?
In

In addition, does the inversive inequality \begin{align*} \|\nabla w\|_{0,\infty,\Omega}\leq Ch^{-1}\|w\|_{0,2,\Omega}, w\in V_h \end{align*} hold?

Can anyone give me some advice Could any one give me a hand?

What's the definition of L^{\infty}-norm for nonconforming finite element?

We know that \begin{align*} \|u\|_{0,\infty,\Omega}={ess \sup}_{x\in \Omega}|u|. \end{align*} Moreover, the nonconforming Crouzeix-Raviart finite element space is \begin{align*} V_h=\{ v|_K\in P_1(K), v ~is~continuous~at~midpoints~of~edges~and~v=0~at~the~midpoints~on~boundary~edges\}. \end{align*} For a function $w\in V_h$, we know that $w\notin H_0^1(\Omega)$. My question is what is the definition of $\|w\|_{0,\infty,\Omega}$ and $\|\nabla w\|_{0,\infty,\Omega}$?
In addition, does the inversive inequality \begin{align*} \|\nabla w\|_{0,\infty,\Omega}\leq Ch^{-1}\|w\|_{0,2,\Omega}, w\in V_h \end{align*} hold?

Can anyone give me some advice Could any one give me a hand?

What's the definition of $L^{\infty}$-norm for nonconforming finite element?

We know that \begin{align*} \|u\|_{0,\infty,\Omega}={\text{ess} \sup}_{x\in \Omega}|u|. \end{align*} Moreover, the nonconforming Crouzeix-Raviart finite element space is \begin{align*} V_h=\{ v|_K\in P_1(K),\ v ~\text{is continuous at the midpoints of edges and } v=0 \text{ at the midpoints on boundary edges}\}. \end{align*} For a function $w\in V_h$, we know that $w\notin H_0^1(\Omega)$. 

My question is what is the definition of $\|w\|_{0,\infty,\Omega}$ and $\|\nabla w\|_{0,\infty,\Omega}$?

In addition, does the inversive inequality \begin{align*} \|\nabla w\|_{0,\infty,\Omega}\leq Ch^{-1}\|w\|_{0,2,\Omega}, w\in V_h \end{align*} hold?

edited body
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Feng Young
Feng Young
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We know that \begin{align*} \|u\|_{0,\infty,\Omega}={ess \sup}_{x\in \Omega}|u|. \end{align*} Moreover, the nonconforming Crouzeix-Raviart finite element space is \begin{align*} V_h=\{ v|_K\in P_1{K}, v ~is~continuous~at~midpoints~of~edges~and~v=0~at~the~midpoints~on~boundary~edges\}. \end{align*}\begin{align*} V_h=\{ v|_K\in P_1(K), v ~is~continuous~at~midpoints~of~edges~and~v=0~at~the~midpoints~on~boundary~edges\}. \end{align*} For a function $w\in V_h$, we know that $w\notin H_0^1(\Omega)$. My question is what is the definition of $\|w\|_{0,\infty,\Omega}$ and $\|\nabla w\|_{0,\infty,\Omega}$?
In addition, does the inversive inequality \begin{align*} \|\nabla w\|_{0,\infty,\Omega}\leq Ch^{-1}\|w\|_{0,2,\Omega}, w\in V_h \end{align*} hold?

Can anyone give me some advice Could any one give me a hand?

We know that \begin{align*} \|u\|_{0,\infty,\Omega}={ess \sup}_{x\in \Omega}|u|. \end{align*} Moreover, the nonconforming Crouzeix-Raviart finite element space is \begin{align*} V_h=\{ v|_K\in P_1{K}, v ~is~continuous~at~midpoints~of~edges~and~v=0~at~the~midpoints~on~boundary~edges\}. \end{align*} For a function $w\in V_h$, we know that $w\notin H_0^1(\Omega)$. My question is what is the definition of $\|w\|_{0,\infty,\Omega}$ and $\|\nabla w\|_{0,\infty,\Omega}$?
In addition, does the inversive inequality \begin{align*} \|\nabla w\|_{0,\infty,\Omega}\leq Ch^{-1}\|w\|_{0,2,\Omega}, w\in V_h \end{align*} hold?

Can anyone give me some advice Could any one give me a hand?

We know that \begin{align*} \|u\|_{0,\infty,\Omega}={ess \sup}_{x\in \Omega}|u|. \end{align*} Moreover, the nonconforming Crouzeix-Raviart finite element space is \begin{align*} V_h=\{ v|_K\in P_1(K), v ~is~continuous~at~midpoints~of~edges~and~v=0~at~the~midpoints~on~boundary~edges\}. \end{align*} For a function $w\in V_h$, we know that $w\notin H_0^1(\Omega)$. My question is what is the definition of $\|w\|_{0,\infty,\Omega}$ and $\|\nabla w\|_{0,\infty,\Omega}$?
In addition, does the inversive inequality \begin{align*} \|\nabla w\|_{0,\infty,\Omega}\leq Ch^{-1}\|w\|_{0,2,\Omega}, w\in V_h \end{align*} hold?

Can anyone give me some advice Could any one give me a hand?

Source Link
Feng Young
Feng Young
  • 113
  • 6

What's the definition of L^{\infty}-norm for nonconforming finite element?

We know that \begin{align*} \|u\|_{0,\infty,\Omega}={ess \sup}_{x\in \Omega}|u|. \end{align*} Moreover, the nonconforming Crouzeix-Raviart finite element space is \begin{align*} V_h=\{ v|_K\in P_1{K}, v ~is~continuous~at~midpoints~of~edges~and~v=0~at~the~midpoints~on~boundary~edges\}. \end{align*} For a function $w\in V_h$, we know that $w\notin H_0^1(\Omega)$. My question is what is the definition of $\|w\|_{0,\infty,\Omega}$ and $\|\nabla w\|_{0,\infty,\Omega}$?
In addition, does the inversive inequality \begin{align*} \|\nabla w\|_{0,\infty,\Omega}\leq Ch^{-1}\|w\|_{0,2,\Omega}, w\in V_h \end{align*} hold?

Can anyone give me some advice Could any one give me a hand?