Mixing
Property of a quasi-uniform triangulation
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I have some type of proof for the inverse inequality:
$|nabla v |_{H^1} le C |v|_{H^1}$
This proof uses the following property for quasi-uniform triangulations:
$ frac{int_{{K}^wedge}{|nabla v^wedge |}^2}{int_{K^wedge}|{v}^wedge |^2} le C$
the wedge refers to the reference triangle. So $K^wedge$ is the reference triangle and $v^wedge$ is the test function on the reference triangle. I'm not sure what type of semi-norm it is, but I assume it's either $L^2$ or $H^1$.
I do not understand why this property is true. Any help appreciated.
finite-element-method triangulation
asked Jan 23 at 13:40
Gilles GGilles G
112
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add a comment |
$begingroup$
I have some type of proof for the inverse inequality:
$|nabla v |_{H^1} le C |v|_{H^1}$
This proof uses the following property for quasi-uniform triangulations:
$ frac{int_{{K}^wedge}{|nabla v^wedge |}^2}{int_{K^wedge}|{v}^wedge |^2} le C$
the wedge refers to the reference triangle. So $K^wedge$ is the reference triangle and $v^wedge$ is the test function on the reference triangle. I'm not sure what type of semi-norm it is, but I assume it's either $L^2$ or $H^1$.
I do not understand why this property is true. Any help appreciated.
finite-element-method triangulation
asked Jan 23 at 13:40
Gilles GGilles G
112
$endgroup$
$begingroup$
Please explain the notation
$endgroup$
– lhf
Jan 23 at 14:52
$begingroup$
I have tried clarifying the notations more...
$endgroup$
– Gilles G
Jan 23 at 15:07
$begingroup$
Inverse inequalities are usually shown for the grid functions. Can you specify what do you mean by a test function? If it’s a polynomial, the estimate is easy. If you mean $H^1$, think of a series of bounded but highly oscillating functions, they will fail the estimate you want. Sorry if I have misunderstood your intentions.
$endgroup$
– VorKir
Feb 3 at 7:40
add a comment |
$begingroup$
I have some type of proof for the inverse inequality:
$|nabla v |_{H^1} le C |v|_{H^1}$
This proof uses the following property for quasi-uniform triangulations:
$ frac{int_{{K}^wedge}{|nabla v^wedge |}^2}{int_{K^wedge}|{v}^wedge |^2} le C$
the wedge refers to the reference triangle. So $K^wedge$ is the reference triangle and $v^wedge$ is the test function on the reference triangle. I'm not sure what type of semi-norm it is, but I assume it's either $L^2$ or $H^1$.
I do not understand why this property is true. Any help appreciated.
finite-element-method triangulation
asked Jan 23 at 13:40
Gilles GGilles G
112
$endgroup$
I have some type of proof for the inverse inequality:
$|nabla v |_{H^1} le C |v|_{H^1}$
This proof uses the following property for quasi-uniform triangulations:
$ frac{int_{{K}^wedge}{|nabla v^wedge |}^2}{int_{K^wedge}|{v}^wedge |^2} le C$
the wedge refers to the reference triangle. So $K^wedge$ is the reference triangle and $v^wedge$ is the test function on the reference triangle. I'm not sure what type of semi-norm it is, but I assume it's either $L^2$ or $H^1$.
I do not understand why this property is true. Any help appreciated.
finite-element-method triangulation
finite-element-method triangulation
asked Jan 23 at 13:40
Gilles GGilles G
112
asked Jan 23 at 13:40
Gilles GGilles G
112
edited Jan 23 at 15:06
Gilles G
asked Jan 23 at 13:40
Gilles GGilles G
112
asked Jan 23 at 13:40
Gilles GGilles G
112
asked Jan 23 at 13:40
Gilles GGilles G
112
112
$begingroup$
Please explain the notation
$endgroup$
– lhf
Jan 23 at 14:52
$begingroup$
I have tried clarifying the notations more...
$endgroup$
– Gilles G
Jan 23 at 15:07
$begingroup$
Inverse inequalities are usually shown for the grid functions. Can you specify what do you mean by a test function? If it’s a polynomial, the estimate is easy. If you mean $H^1$, think of a series of bounded but highly oscillating functions, they will fail the estimate you want. Sorry if I have misunderstood your intentions.
$endgroup$
– VorKir
Feb 3 at 7:40
add a comment |
$begingroup$
Please explain the notation
$endgroup$
– lhf
Jan 23 at 14:52
$begingroup$
I have tried clarifying the notations more...
$endgroup$
– Gilles G
Jan 23 at 15:07
$begingroup$
Inverse inequalities are usually shown for the grid functions. Can you specify what do you mean by a test function? If it’s a polynomial, the estimate is easy. If you mean $H^1$, think of a series of bounded but highly oscillating functions, they will fail the estimate you want. Sorry if I have misunderstood your intentions.
$endgroup$
– VorKir
Feb 3 at 7:40
$begingroup$
Please explain the notation
$endgroup$
– lhf
Jan 23 at 14:52
$begingroup$
Please explain the notation
$endgroup$
– lhf
Jan 23 at 14:52
$begingroup$
I have tried clarifying the notations more...
$endgroup$
– Gilles G
Jan 23 at 15:07
$begingroup$
I have tried clarifying the notations more...
$endgroup$
– Gilles G
Jan 23 at 15:07
$begingroup$
Inverse inequalities are usually shown for the grid functions. Can you specify what do you mean by a test function? If it’s a polynomial, the estimate is easy. If you mean $H^1$, think of a series of bounded but highly oscillating functions, they will fail the estimate you want. Sorry if I have misunderstood your intentions.
$endgroup$
– VorKir
Feb 3 at 7:40
$begingroup$
Inverse inequalities are usually shown for the grid functions. Can you specify what do you mean by a test function? If it’s a polynomial, the estimate is easy. If you mean $H^1$, think of a series of bounded but highly oscillating functions, they will fail the estimate you want. Sorry if I have misunderstood your intentions.
$endgroup$
– VorKir
Feb 3 at 7:40
add a comment |
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$begingroup$
Please explain the notation
$endgroup$
– lhf
Jan 23 at 14:52
$begingroup$
I have tried clarifying the notations more...
$endgroup$
– Gilles G
Jan 23 at 15:07
$begingroup$
Inverse inequalities are usually shown for the grid functions. Can you specify what do you mean by a test function? If it’s a polynomial, the estimate is easy. If you mean $H^1$, think of a series of bounded but highly oscillating functions, they will fail the estimate you want. Sorry if I have misunderstood your intentions.
$endgroup$
– VorKir
Feb 3 at 7:40