1D Finite element method: Function contineously differentiable?
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I m currently reading the book "The finite element method in electromagnetics" written by Jian-Ming Jin.
On page 50 and 51 which I attach as a screenshot below they show for an example functional, that minimizing the functional will lead to the solution of the pde in 3.12.
My question is: In the step 3.8 to 3.9 one did an integration by parts. For this step $alpha frac{dphi}{dx}$ has to be differentiable? If I assume that $alpha$ to be differentiable, then $phi$ has to be differentiable twice.
When I minimize the functinal in practice I discretize space and assume in all discretized space intervals for example linear functions (linear element functions). With these linear functions I stick together my $phi$. But this $phi$ is then of course only contineous, but not twice differentiable!
Does anyone know a solution to this? Maybe it is okay if the discretization is fine enough?
Many thanks in advance.
integration derivatives finite-element-method
asked Jan 6 at 16:07
Jan SEJan SE
114
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add a comment |
$begingroup$
I m currently reading the book "The finite element method in electromagnetics" written by Jian-Ming Jin.
On page 50 and 51 which I attach as a screenshot below they show for an example functional, that minimizing the functional will lead to the solution of the pde in 3.12.
My question is: In the step 3.8 to 3.9 one did an integration by parts. For this step $alpha frac{dphi}{dx}$ has to be differentiable? If I assume that $alpha$ to be differentiable, then $phi$ has to be differentiable twice.
When I minimize the functinal in practice I discretize space and assume in all discretized space intervals for example linear functions (linear element functions). With these linear functions I stick together my $phi$. But this $phi$ is then of course only contineous, but not twice differentiable!
Does anyone know a solution to this? Maybe it is okay if the discretization is fine enough?
Many thanks in advance.
integration derivatives finite-element-method
asked Jan 6 at 16:07
Jan SEJan SE
114
$endgroup$
$begingroup$
Yes, the idea is that in the limit of discretization the solution of the weak (variational) form approaches the solution to the strong (PDE) form. There are many assumptions of course.
$endgroup$
– player100
Jan 6 at 16:15
$begingroup$
Do you have a recommendation where I can read about this in a little bit more detail? It seems that this book goes over these details without even mentioning them...
$endgroup$
– Jan SE
Jan 6 at 17:01
$begingroup$
Are you coming to this from a theoretical or a computational mindset?
$endgroup$
– player100
Jan 6 at 18:26
$begingroup$
In the end I of course want to compute something, but at the moment I try to understand the mathematics.
$endgroup$
– Jan SE
Jan 6 at 23:41
$begingroup$
There are many online resources. Try web.stanford.edu/class/energy281/FiniteElementMethod.pdf There is the much more formal, such as folk.uio.no/kent-and/sommerskole/material/book_20april.pdf
$endgroup$
– player100
Jan 7 at 1:44
add a comment |
$begingroup$
I m currently reading the book "The finite element method in electromagnetics" written by Jian-Ming Jin.
On page 50 and 51 which I attach as a screenshot below they show for an example functional, that minimizing the functional will lead to the solution of the pde in 3.12.
My question is: In the step 3.8 to 3.9 one did an integration by parts. For this step $alpha frac{dphi}{dx}$ has to be differentiable? If I assume that $alpha$ to be differentiable, then $phi$ has to be differentiable twice.
When I minimize the functinal in practice I discretize space and assume in all discretized space intervals for example linear functions (linear element functions). With these linear functions I stick together my $phi$. But this $phi$ is then of course only contineous, but not twice differentiable!
Does anyone know a solution to this? Maybe it is okay if the discretization is fine enough?
Many thanks in advance.
integration derivatives finite-element-method
asked Jan 6 at 16:07
Jan SEJan SE
114
$endgroup$
I m currently reading the book "The finite element method in electromagnetics" written by Jian-Ming Jin.
On page 50 and 51 which I attach as a screenshot below they show for an example functional, that minimizing the functional will lead to the solution of the pde in 3.12.
My question is: In the step 3.8 to 3.9 one did an integration by parts. For this step $alpha frac{dphi}{dx}$ has to be differentiable? If I assume that $alpha$ to be differentiable, then $phi$ has to be differentiable twice.
When I minimize the functinal in practice I discretize space and assume in all discretized space intervals for example linear functions (linear element functions). With these linear functions I stick together my $phi$. But this $phi$ is then of course only contineous, but not twice differentiable!
Does anyone know a solution to this? Maybe it is okay if the discretization is fine enough?
Many thanks in advance.
integration derivatives finite-element-method
integration derivatives finite-element-method
asked Jan 6 at 16:07
Jan SEJan SE
114
asked Jan 6 at 16:07
Jan SEJan SE
114
asked Jan 6 at 16:07
Jan SEJan SE
114
asked Jan 6 at 16:07
Jan SEJan SE
114
asked Jan 6 at 16:07
Jan SEJan SE
114
114
$begingroup$
Yes, the idea is that in the limit of discretization the solution of the weak (variational) form approaches the solution to the strong (PDE) form. There are many assumptions of course.
$endgroup$
– player100
Jan 6 at 16:15
$begingroup$
Do you have a recommendation where I can read about this in a little bit more detail? It seems that this book goes over these details without even mentioning them...
$endgroup$
– Jan SE
Jan 6 at 17:01
$begingroup$
Are you coming to this from a theoretical or a computational mindset?
$endgroup$
– player100
Jan 6 at 18:26
$begingroup$
In the end I of course want to compute something, but at the moment I try to understand the mathematics.
$endgroup$
– Jan SE
Jan 6 at 23:41
$begingroup$
There are many online resources. Try web.stanford.edu/class/energy281/FiniteElementMethod.pdf There is the much more formal, such as folk.uio.no/kent-and/sommerskole/material/book_20april.pdf
$endgroup$
– player100
Jan 7 at 1:44
add a comment |
$begingroup$
Yes, the idea is that in the limit of discretization the solution of the weak (variational) form approaches the solution to the strong (PDE) form. There are many assumptions of course.
$endgroup$
– player100
Jan 6 at 16:15
$begingroup$
Do you have a recommendation where I can read about this in a little bit more detail? It seems that this book goes over these details without even mentioning them...
$endgroup$
– Jan SE
Jan 6 at 17:01
$begingroup$
Are you coming to this from a theoretical or a computational mindset?
$endgroup$
– player100
Jan 6 at 18:26
$begingroup$
In the end I of course want to compute something, but at the moment I try to understand the mathematics.
$endgroup$
– Jan SE
Jan 6 at 23:41
$begingroup$
There are many online resources. Try web.stanford.edu/class/energy281/FiniteElementMethod.pdf There is the much more formal, such as folk.uio.no/kent-and/sommerskole/material/book_20april.pdf
$endgroup$
– player100
Jan 7 at 1:44
$begingroup$
Yes, the idea is that in the limit of discretization the solution of the weak (variational) form approaches the solution to the strong (PDE) form. There are many assumptions of course.
$endgroup$
– player100
Jan 6 at 16:15
$begingroup$
Yes, the idea is that in the limit of discretization the solution of the weak (variational) form approaches the solution to the strong (PDE) form. There are many assumptions of course.
$endgroup$
– player100
Jan 6 at 16:15
$begingroup$
Do you have a recommendation where I can read about this in a little bit more detail? It seems that this book goes over these details without even mentioning them...
$endgroup$
– Jan SE
Jan 6 at 17:01
$begingroup$
Do you have a recommendation where I can read about this in a little bit more detail? It seems that this book goes over these details without even mentioning them...
$endgroup$
– Jan SE
Jan 6 at 17:01
$begingroup$
Are you coming to this from a theoretical or a computational mindset?
$endgroup$
– player100
Jan 6 at 18:26
$begingroup$
Are you coming to this from a theoretical or a computational mindset?
$endgroup$
– player100
Jan 6 at 18:26
$begingroup$
In the end I of course want to compute something, but at the moment I try to understand the mathematics.
$endgroup$
– Jan SE
Jan 6 at 23:41
$begingroup$
In the end I of course want to compute something, but at the moment I try to understand the mathematics.
$endgroup$
– Jan SE
Jan 6 at 23:41
$begingroup$
There are many online resources. Try web.stanford.edu/class/energy281/FiniteElementMethod.pdf There is the much more formal, such as folk.uio.no/kent-and/sommerskole/material/book_20april.pdf
$endgroup$
– player100
Jan 7 at 1:44
$begingroup$
There are many online resources. Try web.stanford.edu/class/energy281/FiniteElementMethod.pdf There is the much more formal, such as folk.uio.no/kent-and/sommerskole/material/book_20april.pdf
$endgroup$
– player100
Jan 7 at 1:44
add a comment |
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$begingroup$
Yes, the idea is that in the limit of discretization the solution of the weak (variational) form approaches the solution to the strong (PDE) form. There are many assumptions of course.
$endgroup$
– player100
Jan 6 at 16:15
$begingroup$
Do you have a recommendation where I can read about this in a little bit more detail? It seems that this book goes over these details without even mentioning them...
$endgroup$
– Jan SE
Jan 6 at 17:01
$begingroup$
Are you coming to this from a theoretical or a computational mindset?
$endgroup$
– player100
Jan 6 at 18:26
$begingroup$
In the end I of course want to compute something, but at the moment I try to understand the mathematics.
$endgroup$
– Jan SE
Jan 6 at 23:41
$begingroup$
There are many online resources. Try web.stanford.edu/class/energy281/FiniteElementMethod.pdf There is the much more formal, such as folk.uio.no/kent-and/sommerskole/material/book_20april.pdf
$endgroup$
– player100
Jan 7 at 1:44