Computing H1 norm numerically












1












$begingroup$


I'm solving a PDE numerically using FDM and Spectral Methods.



I understand how to compute the $L_{2}$, but I dont understand how to compute the $H_{1}$ norm.



What does the $u'$ mean in the below definition?



$E_{H^1}=||u_{h}-u_{true}||=Error_{L^2} +Big(sumlimits_{i=1}^nint_{x_{i-1}}^{x}big({u'_{h}-u'_{true}}big)^2Big)^{1/2}$






numerical-methods norm







share|cite|improve this question









$endgroup$












  • $begingroup$
    $u'$ is the (spatial) derivative of $u$. This comes directly from the definition of the Sobolev $H^1$ norm: en.wikipedia.org/wiki/…
    $endgroup$
    – Erick Wong
    Feb 21 '14 at 3:40










  • $begingroup$
    haha...oops how would you actually compute that in matlab?
    $endgroup$
    – Rambi
    Feb 21 '14 at 4:05
















1












$begingroup$


I'm solving a PDE numerically using FDM and Spectral Methods.



I understand how to compute the $L_{2}$, but I dont understand how to compute the $H_{1}$ norm.



What does the $u'$ mean in the below definition?



$E_{H^1}=||u_{h}-u_{true}||=Error_{L^2} +Big(sumlimits_{i=1}^nint_{x_{i-1}}^{x}big({u'_{h}-u'_{true}}big)^2Big)^{1/2}$






numerical-methods norm







share|cite|improve this question









$endgroup$












  • $begingroup$
    $u'$ is the (spatial) derivative of $u$. This comes directly from the definition of the Sobolev $H^1$ norm: en.wikipedia.org/wiki/…
    $endgroup$
    – Erick Wong
    Feb 21 '14 at 3:40










  • $begingroup$
    haha...oops how would you actually compute that in matlab?
    $endgroup$
    – Rambi
    Feb 21 '14 at 4:05














1












1








1


1



$begingroup$


I'm solving a PDE numerically using FDM and Spectral Methods.



I understand how to compute the $L_{2}$, but I dont understand how to compute the $H_{1}$ norm.



What does the $u'$ mean in the below definition?



$E_{H^1}=||u_{h}-u_{true}||=Error_{L^2} +Big(sumlimits_{i=1}^nint_{x_{i-1}}^{x}big({u'_{h}-u'_{true}}big)^2Big)^{1/2}$






numerical-methods norm







share|cite|improve this question









$endgroup$




I'm solving a PDE numerically using FDM and Spectral Methods.



I understand how to compute the $L_{2}$, but I dont understand how to compute the $H_{1}$ norm.



What does the $u'$ mean in the below definition?



$E_{H^1}=||u_{h}-u_{true}||=Error_{L^2} +Big(sumlimits_{i=1}^nint_{x_{i-1}}^{x}big({u'_{h}-u'_{true}}big)^2Big)^{1/2}$






numerical-methods norm




numerical-methods norm






share|cite|improve this question













share|cite|improve this question











share|cite|improve this question




share|cite|improve this question










asked Feb 21 '14 at 3:24









RambiRambi

185




185












  • $begingroup$
    $u'$ is the (spatial) derivative of $u$. This comes directly from the definition of the Sobolev $H^1$ norm: en.wikipedia.org/wiki/…
    $endgroup$
    – Erick Wong
    Feb 21 '14 at 3:40










  • $begingroup$
    haha...oops how would you actually compute that in matlab?
    $endgroup$
    – Rambi
    Feb 21 '14 at 4:05


















  • $begingroup$
    $u'$ is the (spatial) derivative of $u$. This comes directly from the definition of the Sobolev $H^1$ norm: en.wikipedia.org/wiki/…
    $endgroup$
    – Erick Wong
    Feb 21 '14 at 3:40










  • $begingroup$
    haha...oops how would you actually compute that in matlab?
    $endgroup$
    – Rambi
    Feb 21 '14 at 4:05
















$begingroup$
$u'$ is the (spatial) derivative of $u$. This comes directly from the definition of the Sobolev $H^1$ norm: en.wikipedia.org/wiki/…
$endgroup$
– Erick Wong
Feb 21 '14 at 3:40




$begingroup$
$u'$ is the (spatial) derivative of $u$. This comes directly from the definition of the Sobolev $H^1$ norm: en.wikipedia.org/wiki/…
$endgroup$
– Erick Wong
Feb 21 '14 at 3:40












$begingroup$
haha...oops how would you actually compute that in matlab?
$endgroup$
– Rambi
Feb 21 '14 at 4:05




$begingroup$
haha...oops how would you actually compute that in matlab?
$endgroup$
– Rambi
Feb 21 '14 at 4:05










1 Answer
1






active

oldest

votes


















0












$begingroup$

The finite difference method computes a point-wise approximation of $u_{mathrm{true}}$. You have to do a sort of post-processing of the FDM approximation $u_h$ for which you can compute/approximate its derivative. E.g., in 1D, it is reasonable to reconstruct a $tilde{u}_h$ which is linear on each interval such that $tilde{u}_h(x_i)=u_h(x_i)$ in the point $x_i$ of the discretisation. Such a function you can already differentiate.



For computing the error, I would suggest you to compute the integrals $int_{x_{i-1}}^{x_i}(tilde{u}_h'(x)-u_{mathrm{true}}(x)')^2mathrm{d}x$ on a reasonable discretisation of each interval $[x_{i-1},x_i]$ in order to diminish the quadrature error due to the nonlinearity of $u_{mathrm{true}}$. Of course the same is true also for the $L_2$-norm.






share|cite|improve this answer









$endgroup$













  • $begingroup$
    sounds good! Could you apply the same approach to finding $H_{1}$ numerically for a spectral method with a periodic domain?
    $endgroup$
    – Rambi
    Feb 21 '14 at 14:34










  • $begingroup$
    I'm not very familiar with the spectral method, but I guess that yes except that in that case you don't need to do any post-processing since the solution is already expressed in terms of the spectral basis (if I'm not wrong).
    $endgroup$
    – Algebraic Pavel
    Feb 21 '14 at 14:43











Your Answer





StackExchange.ifUsing("editor", function () {
return StackExchange.using("mathjaxEditing", function () {
StackExchange.MarkdownEditor.creationCallbacks.add(function (editor, postfix) {
StackExchange.mathjaxEditing.prepareWmdForMathJax(editor, postfix, [["$", "$"], ["\\(","\\)"]]);
});
});
}, "mathjax-editing");

StackExchange.ready(function() {
var channelOptions = {
tags: "".split(" "),
id: "69"
};
initTagRenderer("".split(" "), "".split(" "), channelOptions);

StackExchange.using("externalEditor", function() {
// Have to fire editor after snippets, if snippets enabled
if (StackExchange.settings.snippets.snippetsEnabled) {
StackExchange.using("snippets", function() {
createEditor();
});
}
else {
createEditor();
}
});

function createEditor() {
StackExchange.prepareEditor({
heartbeatType: 'answer',
autoActivateHeartbeat: false,
convertImagesToLinks: true,
noModals: true,
showLowRepImageUploadWarning: true,
reputationToPostImages: 10,
bindNavPrevention: true,
postfix: "",
imageUploader: {
brandingHtml: "Powered by u003ca class="icon-imgur-white" href="https://imgur.com/"u003eu003c/au003e",
contentPolicyHtml: "User contributions licensed under u003ca href="https://creativecommons.org/licenses/by-sa/3.0/"u003ecc by-sa 3.0 with attribution requiredu003c/au003e u003ca href="https://stackoverflow.com/legal/content-policy"u003e(content policy)u003c/au003e",
allowUrls: true
},
noCode: true, onDemand: true,
discardSelector: ".discard-answer"
,immediatelyShowMarkdownHelp:true
});


}
});














draft saved

draft discarded


















StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f684389%2fcomputing-h1-norm-numerically%23new-answer', 'question_page');
}
);

Post as a guest















Required, but never shown

























1 Answer
1






active

oldest

votes








1 Answer
1






active

oldest

votes









active

oldest

votes






active

oldest

votes









0












$begingroup$

The finite difference method computes a point-wise approximation of $u_{mathrm{true}}$. You have to do a sort of post-processing of the FDM approximation $u_h$ for which you can compute/approximate its derivative. E.g., in 1D, it is reasonable to reconstruct a $tilde{u}_h$ which is linear on each interval such that $tilde{u}_h(x_i)=u_h(x_i)$ in the point $x_i$ of the discretisation. Such a function you can already differentiate.



For computing the error, I would suggest you to compute the integrals $int_{x_{i-1}}^{x_i}(tilde{u}_h'(x)-u_{mathrm{true}}(x)')^2mathrm{d}x$ on a reasonable discretisation of each interval $[x_{i-1},x_i]$ in order to diminish the quadrature error due to the nonlinearity of $u_{mathrm{true}}$. Of course the same is true also for the $L_2$-norm.






share|cite|improve this answer









$endgroup$













  • $begingroup$
    sounds good! Could you apply the same approach to finding $H_{1}$ numerically for a spectral method with a periodic domain?
    $endgroup$
    – Rambi
    Feb 21 '14 at 14:34










  • $begingroup$
    I'm not very familiar with the spectral method, but I guess that yes except that in that case you don't need to do any post-processing since the solution is already expressed in terms of the spectral basis (if I'm not wrong).
    $endgroup$
    – Algebraic Pavel
    Feb 21 '14 at 14:43
















0












$begingroup$

The finite difference method computes a point-wise approximation of $u_{mathrm{true}}$. You have to do a sort of post-processing of the FDM approximation $u_h$ for which you can compute/approximate its derivative. E.g., in 1D, it is reasonable to reconstruct a $tilde{u}_h$ which is linear on each interval such that $tilde{u}_h(x_i)=u_h(x_i)$ in the point $x_i$ of the discretisation. Such a function you can already differentiate.



For computing the error, I would suggest you to compute the integrals $int_{x_{i-1}}^{x_i}(tilde{u}_h'(x)-u_{mathrm{true}}(x)')^2mathrm{d}x$ on a reasonable discretisation of each interval $[x_{i-1},x_i]$ in order to diminish the quadrature error due to the nonlinearity of $u_{mathrm{true}}$. Of course the same is true also for the $L_2$-norm.






share|cite|improve this answer









$endgroup$













  • $begingroup$
    sounds good! Could you apply the same approach to finding $H_{1}$ numerically for a spectral method with a periodic domain?
    $endgroup$
    – Rambi
    Feb 21 '14 at 14:34










  • $begingroup$
    I'm not very familiar with the spectral method, but I guess that yes except that in that case you don't need to do any post-processing since the solution is already expressed in terms of the spectral basis (if I'm not wrong).
    $endgroup$
    – Algebraic Pavel
    Feb 21 '14 at 14:43














0












0








0





$begingroup$

The finite difference method computes a point-wise approximation of $u_{mathrm{true}}$. You have to do a sort of post-processing of the FDM approximation $u_h$ for which you can compute/approximate its derivative. E.g., in 1D, it is reasonable to reconstruct a $tilde{u}_h$ which is linear on each interval such that $tilde{u}_h(x_i)=u_h(x_i)$ in the point $x_i$ of the discretisation. Such a function you can already differentiate.



For computing the error, I would suggest you to compute the integrals $int_{x_{i-1}}^{x_i}(tilde{u}_h'(x)-u_{mathrm{true}}(x)')^2mathrm{d}x$ on a reasonable discretisation of each interval $[x_{i-1},x_i]$ in order to diminish the quadrature error due to the nonlinearity of $u_{mathrm{true}}$. Of course the same is true also for the $L_2$-norm.






share|cite|improve this answer









$endgroup$



The finite difference method computes a point-wise approximation of $u_{mathrm{true}}$. You have to do a sort of post-processing of the FDM approximation $u_h$ for which you can compute/approximate its derivative. E.g., in 1D, it is reasonable to reconstruct a $tilde{u}_h$ which is linear on each interval such that $tilde{u}_h(x_i)=u_h(x_i)$ in the point $x_i$ of the discretisation. Such a function you can already differentiate.



For computing the error, I would suggest you to compute the integrals $int_{x_{i-1}}^{x_i}(tilde{u}_h'(x)-u_{mathrm{true}}(x)')^2mathrm{d}x$ on a reasonable discretisation of each interval $[x_{i-1},x_i]$ in order to diminish the quadrature error due to the nonlinearity of $u_{mathrm{true}}$. Of course the same is true also for the $L_2$-norm.







share|cite|improve this answer












share|cite|improve this answer



share|cite|improve this answer










answered Feb 21 '14 at 13:07









Algebraic PavelAlgebraic Pavel

16.3k31840




16.3k31840












  • $begingroup$
    sounds good! Could you apply the same approach to finding $H_{1}$ numerically for a spectral method with a periodic domain?
    $endgroup$
    – Rambi
    Feb 21 '14 at 14:34










  • $begingroup$
    I'm not very familiar with the spectral method, but I guess that yes except that in that case you don't need to do any post-processing since the solution is already expressed in terms of the spectral basis (if I'm not wrong).
    $endgroup$
    – Algebraic Pavel
    Feb 21 '14 at 14:43


















  • $begingroup$
    sounds good! Could you apply the same approach to finding $H_{1}$ numerically for a spectral method with a periodic domain?
    $endgroup$
    – Rambi
    Feb 21 '14 at 14:34










  • $begingroup$
    I'm not very familiar with the spectral method, but I guess that yes except that in that case you don't need to do any post-processing since the solution is already expressed in terms of the spectral basis (if I'm not wrong).
    $endgroup$
    – Algebraic Pavel
    Feb 21 '14 at 14:43
















$begingroup$
sounds good! Could you apply the same approach to finding $H_{1}$ numerically for a spectral method with a periodic domain?
$endgroup$
– Rambi
Feb 21 '14 at 14:34




$begingroup$
sounds good! Could you apply the same approach to finding $H_{1}$ numerically for a spectral method with a periodic domain?
$endgroup$
– Rambi
Feb 21 '14 at 14:34












$begingroup$
I'm not very familiar with the spectral method, but I guess that yes except that in that case you don't need to do any post-processing since the solution is already expressed in terms of the spectral basis (if I'm not wrong).
$endgroup$
– Algebraic Pavel
Feb 21 '14 at 14:43




$begingroup$
I'm not very familiar with the spectral method, but I guess that yes except that in that case you don't need to do any post-processing since the solution is already expressed in terms of the spectral basis (if I'm not wrong).
$endgroup$
– Algebraic Pavel
Feb 21 '14 at 14:43


















draft saved

draft discarded




















































Thanks for contributing an answer to Mathematics Stack Exchange!


  • Please be sure to answer the question. Provide details and share your research!

But avoid …



  • Asking for help, clarification, or responding to other answers.

  • Making statements based on opinion; back them up with references or personal experience.


Use MathJax to format equations. MathJax reference.


To learn more, see our tips on writing great answers.




draft saved


draft discarded














StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f684389%2fcomputing-h1-norm-numerically%23new-answer', 'question_page');
}
);

Post as a guest















Required, but never shown





















































Required, but never shown














Required, but never shown












Required, but never shown







Required, but never shown

































Required, but never shown














Required, but never shown












Required, but never shown







Required, but never shown







-

Popular posts from this blog

To store a contact into the json file from server.js file using a class in NodeJS

Image
0 I have created three files -contactPresentation.js, contactService.js, contacts.json. I will run the contactService.js file. And it will navigate to contactPresentation.js file that will only choose an option to either add or display a file. If add(), then add the data into add of service file.And the data will be added to the contacts.json file . And if display(), then display the display() of service file. And the display will take the data from contacts.json file. However, i know my code is flawed. Please correct it . I am also getting an error - this.contacts.push(new ContactService(userName, contactNumber, emailId)); ^ TypeError: Cannot read property 'push' of undefined at ContactService.add (e:NodeJSProjectcontactManager2contactService.js:57:23) at new C
Read more

Redirect URL with Chrome Remote Debugging Android Devices

Image
1 How can i redirect an URL with Chrome Remote Debugging Android Devices? like Requestly Chrome extension on desktop. google-chrome debugging url redirect testing share | improve this question edited Nov 24 '18 at 8:16 sachinjain024 14.7k 23 73 129 asked Nov 23 '18 at 16:09 Momen Momen 91 10
Read more

Dieringhausen

Image
Dieringhausen Stadt Gummersbach 50.982777777778 7.5316666666667 165 Koordinaten: 50° 58′ 58″  N , 7° 31′ 54″  O Höhe: 165  (160–245)  m Einwohner: 5055  (30. Jun. 2016) Postleitzahl: 51645 Vorwahl: 02261 Lage von Dieringhausen in Gummersbach Im Hohl ca. 1927–1930 Ansicht von Südwesten bei Bünghausen Dieringhausen (im örtlichen Dialekt Dierkesen ) ist ein Ortsteil von Gummersbach im Oberbergischen Kreis, Regierungsbezirk Köln, Nordrhein-Westfalen (Deutschland). Inhaltsverzeichnis 1 Geographie 2 Geschichte 3 Kultur 3.1 Sehenswürdigkeiten 3.2 Regelmäßige Veranstaltungen 4 Infrastruktur und Wirtschaft 4.1 Verkehr 4.1.1 Schienen- und Busverkehr 4.1.2 Straßen 4.2 Öffentliche Einrichtungen 4.2.1 Schulen und Bildungseinrichtungen 4.2.2 Kirchliche Einrichtungen 5 Persönlichkeiten 5.1 In Dieringhausen geboren 5.2 In Dieringhausen gelebt 6 Einzelnachweise 7
Read more