Does Picard iteration affect the convergence order of a numerical scheme?
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I have a common nonlinear differential equation, for example, the one in Stokes problem: Find $u$ (velocity) and $p$ (pressure) such that
$$nablacdot(-mu(u)nabla u+p,I)=fqquadtextrm{ in }Omega$$
$$nablacdot u=0qquadtextrm{ in }Omega$$
$$textrm{plus boundary conditions}$$
where $mu$ is a nonlinear function depending on $u$ and $f$ is a known data.
Applying the Picard iteration I obtain the (lineal) problem: Find $u^{j+1}$ and $p^{j+1}$ such that
$$nablacdot(-mu(u^j)nabla u^{j+1}+p^{j+1},I)=fqquadtextrm{ in }Omega$$
$$nablacdot u^{j+1}=0qquadtextrm{ in }Omega$$
$$textrm{plus boundary conditions}$$
where $u^j$ is the velocity in the before step (a known data).
From here, I can apply a linear finite element scheme to solve the last problem (the linear problem).
If the finite element scheme has order $k$, have the nonlinear scheme also order $k$? Or does the Picard iteration affect the order of the method?
numerical-methods nonlinear-system finite-element-method
edited Nov 23 at 17:27
Daniele Tampieri
1,5791619
asked Nov 23 at 16:52
yemino
2631314
add a comment |
up vote
0
down vote
favorite
I have a common nonlinear differential equation, for example, the one in Stokes problem: Find $u$ (velocity) and $p$ (pressure) such that
$$nablacdot(-mu(u)nabla u+p,I)=fqquadtextrm{ in }Omega$$
$$nablacdot u=0qquadtextrm{ in }Omega$$
$$textrm{plus boundary conditions}$$
where $mu$ is a nonlinear function depending on $u$ and $f$ is a known data.
Applying the Picard iteration I obtain the (lineal) problem: Find $u^{j+1}$ and $p^{j+1}$ such that
$$nablacdot(-mu(u^j)nabla u^{j+1}+p^{j+1},I)=fqquadtextrm{ in }Omega$$
$$nablacdot u^{j+1}=0qquadtextrm{ in }Omega$$
$$textrm{plus boundary conditions}$$
where $u^j$ is the velocity in the before step (a known data).
From here, I can apply a linear finite element scheme to solve the last problem (the linear problem).
If the finite element scheme has order $k$, have the nonlinear scheme also order $k$? Or does the Picard iteration affect the order of the method?
numerical-methods nonlinear-system finite-element-method
edited Nov 23 at 17:27
Daniele Tampieri
1,5791619
asked Nov 23 at 16:52
yemino
2631314
add a comment |
up vote
0
down vote
favorite
up vote
0
down vote
favorite
I have a common nonlinear differential equation, for example, the one in Stokes problem: Find $u$ (velocity) and $p$ (pressure) such that
$$nablacdot(-mu(u)nabla u+p,I)=fqquadtextrm{ in }Omega$$
$$nablacdot u=0qquadtextrm{ in }Omega$$
$$textrm{plus boundary conditions}$$
where $mu$ is a nonlinear function depending on $u$ and $f$ is a known data.
Applying the Picard iteration I obtain the (lineal) problem: Find $u^{j+1}$ and $p^{j+1}$ such that
$$nablacdot(-mu(u^j)nabla u^{j+1}+p^{j+1},I)=fqquadtextrm{ in }Omega$$
$$nablacdot u^{j+1}=0qquadtextrm{ in }Omega$$
$$textrm{plus boundary conditions}$$
where $u^j$ is the velocity in the before step (a known data).
From here, I can apply a linear finite element scheme to solve the last problem (the linear problem).
If the finite element scheme has order $k$, have the nonlinear scheme also order $k$? Or does the Picard iteration affect the order of the method?
numerical-methods nonlinear-system finite-element-method
edited Nov 23 at 17:27
Daniele Tampieri
1,5791619
asked Nov 23 at 16:52
yemino
2631314
I have a common nonlinear differential equation, for example, the one in Stokes problem: Find $u$ (velocity) and $p$ (pressure) such that
$$nablacdot(-mu(u)nabla u+p,I)=fqquadtextrm{ in }Omega$$
$$nablacdot u=0qquadtextrm{ in }Omega$$
$$textrm{plus boundary conditions}$$
where $mu$ is a nonlinear function depending on $u$ and $f$ is a known data.
Applying the Picard iteration I obtain the (lineal) problem: Find $u^{j+1}$ and $p^{j+1}$ such that
$$nablacdot(-mu(u^j)nabla u^{j+1}+p^{j+1},I)=fqquadtextrm{ in }Omega$$
$$nablacdot u^{j+1}=0qquadtextrm{ in }Omega$$
$$textrm{plus boundary conditions}$$
where $u^j$ is the velocity in the before step (a known data).
From here, I can apply a linear finite element scheme to solve the last problem (the linear problem).
If the finite element scheme has order $k$, have the nonlinear scheme also order $k$? Or does the Picard iteration affect the order of the method?
numerical-methods nonlinear-system finite-element-method
numerical-methods nonlinear-system finite-element-method
edited Nov 23 at 17:27
Daniele Tampieri
1,5791619
asked Nov 23 at 16:52
yemino
2631314
edited Nov 23 at 17:27
Daniele Tampieri
1,5791619
asked Nov 23 at 16:52
yemino
2631314
edited Nov 23 at 17:27
Daniele Tampieri
1,5791619
edited Nov 23 at 17:27
Daniele Tampieri
1,5791619
edited Nov 23 at 17:27
Daniele Tampieri
1,5791619
1,5791619
asked Nov 23 at 16:52
yemino
2631314
asked Nov 23 at 16:52
yemino
2631314
asked Nov 23 at 16:52
yemino
2631314
2631314
add a comment |
add a comment |
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