Mixing
Finite Element Analysis
$begingroup$
We have the follwing diffusion-convection-reaction equation PDE and we want to solve it using Finite Elemrnts:
$$ -nabla.(k(x)nabla u)+overrightarrow {b}nabla u+ru=f in Omega $$
$$knabla u.overrightarrow n=g_1 on Gamma_1$$
$$knabla u.overrightarrow n +alpha u=g_2 on Gamma_2$$
$$u=g_3=0 on Gamma_3$$
$$partialOmega=Gamma_1 cupGamma_2 cupGamma_3$$
Let
$a(.,.):VxX rightarrow R$ s.t :
$$ a(u,v)=int_{Omega}knabla u nabla v dx + int_{Omega}overrightarrow bnabla u v dx+int_{Omega}ruv dx +int_{Gamma_2}alpha uv ds $$
$L(.):Vrightarrow R$ s.t:
$$L(v)=int_{Omega} fv dx + int_{Gamma_1}g_1v dsint_{Gamma_2} g_2v ds+$$
$$V_{0,Gamma_3}:={v in H^{1}/v on Gamma_3=0}$$
Our problem now is find $$u in V_{0,Gamma_3}$$ s.t:
$$a(u,v)=L(v)$$
I have struggled how to prove the coersivity and continuity of$ a(u,v)$ and the continuity of $L(v)$.
Could someone help me out as I dont have strong background in Functional Analysis
numerical-methods
asked Jan 13 at 18:22
FarhanFarhan
547
$endgroup$
add a comment |
$begingroup$
We have the follwing diffusion-convection-reaction equation PDE and we want to solve it using Finite Elemrnts:
$$ -nabla.(k(x)nabla u)+overrightarrow {b}nabla u+ru=f in Omega $$
$$knabla u.overrightarrow n=g_1 on Gamma_1$$
$$knabla u.overrightarrow n +alpha u=g_2 on Gamma_2$$
$$u=g_3=0 on Gamma_3$$
$$partialOmega=Gamma_1 cupGamma_2 cupGamma_3$$
Let
$a(.,.):VxX rightarrow R$ s.t :
$$ a(u,v)=int_{Omega}knabla u nabla v dx + int_{Omega}overrightarrow bnabla u v dx+int_{Omega}ruv dx +int_{Gamma_2}alpha uv ds $$
$L(.):Vrightarrow R$ s.t:
$$L(v)=int_{Omega} fv dx + int_{Gamma_1}g_1v dsint_{Gamma_2} g_2v ds+$$
$$V_{0,Gamma_3}:={v in H^{1}/v on Gamma_3=0}$$
Our problem now is find $$u in V_{0,Gamma_3}$$ s.t:
$$a(u,v)=L(v)$$
I have struggled how to prove the coersivity and continuity of$ a(u,v)$ and the continuity of $L(v)$.
Could someone help me out as I dont have strong background in Functional Analysis
numerical-methods
asked Jan 13 at 18:22
FarhanFarhan
547
$endgroup$
add a comment |
$begingroup$
We have the follwing diffusion-convection-reaction equation PDE and we want to solve it using Finite Elemrnts:
$$ -nabla.(k(x)nabla u)+overrightarrow {b}nabla u+ru=f in Omega $$
$$knabla u.overrightarrow n=g_1 on Gamma_1$$
$$knabla u.overrightarrow n +alpha u=g_2 on Gamma_2$$
$$u=g_3=0 on Gamma_3$$
$$partialOmega=Gamma_1 cupGamma_2 cupGamma_3$$
Let
$a(.,.):VxX rightarrow R$ s.t :
$$ a(u,v)=int_{Omega}knabla u nabla v dx + int_{Omega}overrightarrow bnabla u v dx+int_{Omega}ruv dx +int_{Gamma_2}alpha uv ds $$
$L(.):Vrightarrow R$ s.t:
$$L(v)=int_{Omega} fv dx + int_{Gamma_1}g_1v dsint_{Gamma_2} g_2v ds+$$
$$V_{0,Gamma_3}:={v in H^{1}/v on Gamma_3=0}$$
Our problem now is find $$u in V_{0,Gamma_3}$$ s.t:
$$a(u,v)=L(v)$$
I have struggled how to prove the coersivity and continuity of$ a(u,v)$ and the continuity of $L(v)$.
Could someone help me out as I dont have strong background in Functional Analysis
numerical-methods
asked Jan 13 at 18:22
FarhanFarhan
547
$endgroup$
We have the follwing diffusion-convection-reaction equation PDE and we want to solve it using Finite Elemrnts:
$$ -nabla.(k(x)nabla u)+overrightarrow {b}nabla u+ru=f in Omega $$
$$knabla u.overrightarrow n=g_1 on Gamma_1$$
$$knabla u.overrightarrow n +alpha u=g_2 on Gamma_2$$
$$u=g_3=0 on Gamma_3$$
$$partialOmega=Gamma_1 cupGamma_2 cupGamma_3$$
Let
$a(.,.):VxX rightarrow R$ s.t :
$$ a(u,v)=int_{Omega}knabla u nabla v dx + int_{Omega}overrightarrow bnabla u v dx+int_{Omega}ruv dx +int_{Gamma_2}alpha uv ds $$
$L(.):Vrightarrow R$ s.t:
$$L(v)=int_{Omega} fv dx + int_{Gamma_1}g_1v dsint_{Gamma_2} g_2v ds+$$
$$V_{0,Gamma_3}:={v in H^{1}/v on Gamma_3=0}$$
Our problem now is find $$u in V_{0,Gamma_3}$$ s.t:
$$a(u,v)=L(v)$$
I have struggled how to prove the coersivity and continuity of$ a(u,v)$ and the continuity of $L(v)$.
Could someone help me out as I dont have strong background in Functional Analysis
numerical-methods
numerical-methods
asked Jan 13 at 18:22
FarhanFarhan
547
asked Jan 13 at 18:22
FarhanFarhan
547
edited Jan 13 at 20:32
Farhan
asked Jan 13 at 18:22
FarhanFarhan
547
asked Jan 13 at 18:22
FarhanFarhan
547
asked Jan 13 at 18:22
FarhanFarhan
547
547
add a comment |
add a comment |
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